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Home , Natural logarithm

I Domain = (0, ∞)

I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1. d(ln x) 1 I dx = x I The graph of y = ln x is increasing, continuous and concave down on the interval (0, ∞).

I The f (x) = ln x is a one-to-one function

I Since f (x) = ln x is a one-to-one function, there is a unique number, e, with the property that ln e = 1.

Natural Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the of 1/x Differentiation and integration Logarithmic differentiation summaries Definition and properties of ln(x).

We define a new function Z x 1 ln x = dt, x > 0. 1 t This function is called the natural logarithm. We derive a number of properties of this new function:

Annette Pilkington Natural Logarithm and Natural Exponential I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1. d(ln x) 1 I dx = x I The graph of y = ln x is increasing, continuous and concave down on the interval (0, ∞).

I The function f (x) = ln x is a one-to-one function

I Since f (x) = ln x is a one-to-one function, there is a unique number, e, with the property that ln e = 1.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Definition and properties of ln(x).

We define a new function Z x 1 ln x = dt, x > 0. 1 t This function is called the natural logarithm. We derive a number of properties of this new function:

I Domain = (0, ∞)

Annette Pilkington Natural Logarithm and Natural Exponential d(ln x) 1 I dx = x I The graph of y = ln x is increasing, continuous and concave down on the interval (0, ∞).

I The function f (x) = ln x is a one-to-one function

I Since f (x) = ln x is a one-to-one function, there is a unique number, e, with the property that ln e = 1.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Definition and properties of ln(x).

We define a new function Z x 1 ln x = dt, x > 0. 1 t This function is called the natural logarithm. We derive a number of properties of this new function:

I Domain = (0, ∞)

I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1.

Annette Pilkington Natural Logarithm and Natural Exponential I The graph of y = ln x is increasing, continuous and concave down on the interval (0, ∞).

I The function f (x) = ln x is a one-to-one function

I Since f (x) = ln x is a one-to-one function, there is a unique number, e, with the property that ln e = 1.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Definition and properties of ln(x).

We define a new function Z x 1 ln x = dt, x > 0. 1 t This function is called the natural logarithm. We derive a number of properties of this new function:

I Domain = (0, ∞)

I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1. d(ln x) 1 I dx = x

Annette Pilkington Natural Logarithm and Natural Exponential I The function f (x) = ln x is a one-to-one function

I Since f (x) = ln x is a one-to-one function, there is a unique number, e, with the property that ln e = 1.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Definition and properties of ln(x).

We define a new function Z x 1 ln x = dt, x > 0. 1 t This function is called the natural logarithm. We derive a number of properties of this new function:

I Domain = (0, ∞)

I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1. d(ln x) 1 I dx = x I The graph of y = ln x is increasing, continuous and concave down on the interval (0, ∞).

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Definition and properties of ln(x).

We define a new function Z x 1 ln x = dt, x > 0. 1 t This function is called the natural logarithm. We derive a number of properties of this new function:

I Domain = (0, ∞)

I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1. d(ln x) 1 I dx = x I The graph of y = ln x is increasing, continuous and concave down on the interval (0, ∞).

I The function f (x) = ln x is a one-to-one function

I Since f (x) = ln x is a one-to-one function, there is a unique number, e, with the property that ln e = 1.

Annette Pilkington Natural Logarithm and Natural Exponential 1.5

1.0

0.5

1 2 e 3 4 5

-0.5

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Graph of ln(x).

Using the information derived above, we can sketch a graph of the natural logarithm

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Graph of ln(x).

Using the information derived above, we can sketch a graph of the natural logarithm

1.5

1.0

0.5

1 2 e 3 4 5

-0.5

Annette Pilkington Natural Logarithm and Natural Exponential IIlnI 1 = 0

I ln(ab) = ln a + ln b r I ln a = r ln a

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Rules of

We also derived the following algebraic properties of our new function by comparing . We can use these algebraic rules to simplify the natural logarithm of products and quotients:

Annette Pilkington Natural Logarithm and Natural Exponential I ln(ab) = ln a + ln b r I ln a = r ln a

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Rules of Logarithms

We also derived the following algebraic properties of our new function by comparing derivatives. We can use these algebraic rules to simplify the natural logarithm of products and quotients:

I ln 1 = 0

Annette Pilkington Natural Logarithm and Natural Exponential r I ln a = r ln a

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Rules of Logarithms

We also derived the following algebraic properties of our new function by comparing derivatives. We can use these algebraic rules to simplify the natural logarithm of products and quotients:

I ln 1 = 0

I ln(ab) = ln a + ln b

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Rules of Logarithms

We also derived the following algebraic properties of our new function by comparing derivatives. We can use these algebraic rules to simplify the natural logarithm of products and quotients:

I ln 1 = 0

I ln(ab) = ln a + ln b r I ln a = r ln a

Annette Pilkington Natural Logarithm and Natural Exponential I From the definition of ln(x), we have

2 Z e 1 dt = ln(e2) 1 t

I = 2 ln e = 2.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Example using definition of e and rule 3

2 R e 1 Example Evaluate 1 t dt

Annette Pilkington Natural Logarithm and Natural Exponential I = 2 ln e = 2.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Example using definition of e and rule 3

2 R e 1 Example Evaluate 1 t dt I From the definition of ln(x), we have

2 Z e 1 dt = ln(e2) 1 t

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Example using definition of e and rule 3

2 R e 1 Example Evaluate 1 t dt I From the definition of ln(x), we have

2 Z e 1 dt = ln(e2) 1 t

I = 2 ln e = 2.

Annette Pilkington Natural Logarithm and Natural Exponential 1 I As x → ∞, we have x2+1 → 0 1 I Therfore as x → ∞, ln( x2+1 ) → −∞ [= limu→0 ln(u)]

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Limits at ∞ and 0.

We can use the rules of logarithms given above to derive the following information about limits.

lim ln x = ∞, lim ln x = −∞. x→∞ x→0

(see notes for a proof) 1 Example Find the limx→∞ ln( x2+1 ).

Annette Pilkington Natural Logarithm and Natural Exponential 1 I Therfore as x → ∞, ln( x2+1 ) → −∞ [= limu→0 ln(u)]

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Limits at ∞ and 0.

We can use the rules of logarithms given above to derive the following information about limits.

lim ln x = ∞, lim ln x = −∞. x→∞ x→0

(see notes for a proof) 1 Example Find the limit limx→∞ ln( x2+1 ). 1 I As x → ∞, we have x2+1 → 0

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Limits at ∞ and 0.

We can use the rules of logarithms given above to derive the following information about limits.

lim ln x = ∞, lim ln x = −∞. x→∞ x→0

(see notes for a proof) 1 Example Find the limit limx→∞ ln( x2+1 ). 1 I As x → ∞, we have x2+1 → 0 1 I Therfore as x → ∞, ln( x2+1 ) → −∞ [= limu→0 ln(u)]

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries ln |x|

We can extend the applications of the natural logarithm function by composing it with the absolute value function. We have :  ln x x > 0 ln |x| = ln(−x) x < 0 This is an even function with graph

3

2

1

-20 -10 10 20

-1 We have ln|x| is also an antiderivative of 1/x with a larger domain than ln(x).

d 1 Z 1 (ln |x|) = and dx = ln |x| + dx x x

Annette Pilkington Natural Logarithm and Natural Exponential I Using the , we have d 1 d ln | sin x| = sin x dx sin x dx I cos x = sin x

d “ 1 ” I We can simplify this to finding ln |x − 1| , since √ dx 3 ln | 3 x − 1| = ln |x − 1|1/3 I d 1 1 1 d 1 ln |x − 1| = (x − 1) = dx 3 3 (x − 1) dx 3(x − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Chain Rule

d 1 d g 0(x) (ln |x|) = and (ln |g(x)|) = dx x dx g(x)

Example Differentiate ln | sin x|.

√ Example Differentiate ln | 3 x − 1|.

Annette Pilkington Natural Logarithm and Natural Exponential I cos x = sin x

d “ 1 ” I We can simplify this to finding ln |x − 1| , since √ dx 3 ln | 3 x − 1| = ln |x − 1|1/3 I d 1 1 1 d 1 ln |x − 1| = (x − 1) = dx 3 3 (x − 1) dx 3(x − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Chain Rule

d 1 d g 0(x) (ln |x|) = and (ln |g(x)|) = dx x dx g(x)

Example Differentiate ln | sin x|. I Using the chain rule, we have d 1 d ln | sin x| = sin x dx sin x dx

√ Example Differentiate ln | 3 x − 1|.

Annette Pilkington Natural Logarithm and Natural Exponential d “ 1 ” I We can simplify this to finding ln |x − 1| , since √ dx 3 ln | 3 x − 1| = ln |x − 1|1/3 I d 1 1 1 d 1 ln |x − 1| = (x − 1) = dx 3 3 (x − 1) dx 3(x − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Chain Rule

d 1 d g 0(x) (ln |x|) = and (ln |g(x)|) = dx x dx g(x)

Example Differentiate ln | sin x|. I Using the chain rule, we have d 1 d ln | sin x| = sin x dx sin x dx I cos x = sin x √ Example Differentiate ln | 3 x − 1|.

Annette Pilkington Natural Logarithm and Natural Exponential I d 1 1 1 d 1 ln |x − 1| = (x − 1) = dx 3 3 (x − 1) dx 3(x − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Chain Rule

d 1 d g 0(x) (ln |x|) = and (ln |g(x)|) = dx x dx g(x)

Example Differentiate ln | sin x|. I Using the chain rule, we have d 1 d ln | sin x| = sin x dx sin x dx I cos x = sin x √ Example Differentiate ln | 3 x − 1|. d “ 1 ” I We can simplify this to finding ln |x − 1| , since √ dx 3 ln | 3 x − 1| = ln |x − 1|1/3

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Chain Rule

d 1 d g 0(x) (ln |x|) = and (ln |g(x)|) = dx x dx g(x)

Example Differentiate ln | sin x|. I Using the chain rule, we have d 1 d ln | sin x| = sin x dx sin x dx I cos x = sin x √ Example Differentiate ln | 3 x − 1|. d “ 1 ” I We can simplify this to finding ln |x − 1| , since √ dx 3 ln | 3 x − 1| = ln |x − 1|1/3 I d 1 1 1 d 1 ln |x − 1| = (x − 1) = dx 3 3 (x − 1) dx 3(x − 1)

Annette Pilkington Natural Logarithm and Natural Exponential 2 I Using substitution, we let u = 3 − x . du du = −2x dx, x dx = , −2

I Z x Z 1 dx = du 3 − x 2 −2(u)

I −1 −1 = ln |u| + C = ln |3 − x 2| + C 2 2

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Substitution

Z 1 Z g 0(x) dx = ln |x| + C and dx = ln |g(x)| + C x g(x)

Example Find the Z x dx. 3 − x 2

Annette Pilkington Natural Logarithm and Natural Exponential I Z x Z 1 dx = du 3 − x 2 −2(u)

I −1 −1 = ln |u| + C = ln |3 − x 2| + C 2 2

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Substitution

Z 1 Z g 0(x) dx = ln |x| + C and dx = ln |g(x)| + C x g(x)

Example Find the integral Z x dx. 3 − x 2

2 I Using substitution, we let u = 3 − x . du du = −2x dx, x dx = , −2

Annette Pilkington Natural Logarithm and Natural Exponential I −1 −1 = ln |u| + C = ln |3 − x 2| + C 2 2

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Substitution

Z 1 Z g 0(x) dx = ln |x| + C and dx = ln |g(x)| + C x g(x)

Example Find the integral Z x dx. 3 − x 2

2 I Using substitution, we let u = 3 − x . du du = −2x dx, x dx = , −2

I Z x Z 1 dx = du 3 − x 2 −2(u)

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Using Substitution

Z 1 Z g 0(x) dx = ln |x| + C and dx = ln |g(x)| + C x g(x)

Example Find the integral Z x dx. 3 − x 2

2 I Using substitution, we let u = 3 − x . du du = −2x dx, x dx = , −2

I Z x Z 1 dx = du 3 − x 2 −2(u)

I −1 −1 = ln |u| + C = ln |3 − x 2| + C 2 2

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Logarithmic differentiation

To differentiate y = f (x), it is often easier to use logarithmic differentiation : 1. Take the natural logarithm of both sides to get ln y = ln(f (x)). 1 dy d 2. Differentiate with respect to x to get y dx = dx ln(f (x)) dy d d 3. We get dx = y dx ln(f (x)) = f (x) dx ln(f (x)).

Annette Pilkington Natural Logarithm and Natural Exponential I We take the natural logarithm of both sides to get r x 2 + 1 ln y = ln 4 x 2 − 1 I Using the rules of logarithms to expand the R.H.S. we get 1 x 2 + 1 1 h i 1 1 ln y = ln = ln(x 2 +1)−ln(x 2 −1) = ln(x 2 +1)− ln(x 2 −1) 4 x 2 − 1 4 4 4 I Differentiating both sides with respect to x, we get 1 dy 1 2x 1 2x x x = · − · = − y dx 4 (x 2 + 1) 4 (x 2 − 1) 2(x 2 + 1) 2(x 2 − 1)

I Multiplying both sides by y, we get dy h x x i = y − dx 2(x 2 + 1) 2(x 2 − 1)

I Convertingy to a function of x, we get r " # dy x 2 + 1 x x = 4 − dx x 2 − 1 2(x 2 + 1) 2(x 2 − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Logarithmic differentiation q 4 x2+1 Example Find the of y = x2−1 .

Annette Pilkington Natural Logarithm and Natural Exponential I Using the rules of logarithms to expand the R.H.S. we get 1 x 2 + 1 1 h i 1 1 ln y = ln = ln(x 2 +1)−ln(x 2 −1) = ln(x 2 +1)− ln(x 2 −1) 4 x 2 − 1 4 4 4 I Differentiating both sides with respect to x, we get 1 dy 1 2x 1 2x x x = · − · = − y dx 4 (x 2 + 1) 4 (x 2 − 1) 2(x 2 + 1) 2(x 2 − 1)

I Multiplying both sides by y, we get dy h x x i = y − dx 2(x 2 + 1) 2(x 2 − 1)

I Convertingy to a function of x, we get r " # dy x 2 + 1 x x = 4 − dx x 2 − 1 2(x 2 + 1) 2(x 2 − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Logarithmic differentiation q 4 x2+1 Example Find the derivative of y = x2−1 . I We take the natural logarithm of both sides to get r x 2 + 1 ln y = ln 4 x 2 − 1

Annette Pilkington Natural Logarithm and Natural Exponential I Differentiating both sides with respect to x, we get 1 dy 1 2x 1 2x x x = · − · = − y dx 4 (x 2 + 1) 4 (x 2 − 1) 2(x 2 + 1) 2(x 2 − 1)

I Multiplying both sides by y, we get dy h x x i = y − dx 2(x 2 + 1) 2(x 2 − 1)

I Convertingy to a function of x, we get r " # dy x 2 + 1 x x = 4 − dx x 2 − 1 2(x 2 + 1) 2(x 2 − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Logarithmic differentiation q 4 x2+1 Example Find the derivative of y = x2−1 . I We take the natural logarithm of both sides to get r x 2 + 1 ln y = ln 4 x 2 − 1 I Using the rules of logarithms to expand the R.H.S. we get 1 x 2 + 1 1 h i 1 1 ln y = ln = ln(x 2 +1)−ln(x 2 −1) = ln(x 2 +1)− ln(x 2 −1) 4 x 2 − 1 4 4 4

Annette Pilkington Natural Logarithm and Natural Exponential I Multiplying both sides by y, we get dy h x x i = y − dx 2(x 2 + 1) 2(x 2 − 1)

I Convertingy to a function of x, we get r " # dy x 2 + 1 x x = 4 − dx x 2 − 1 2(x 2 + 1) 2(x 2 − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Logarithmic differentiation q 4 x2+1 Example Find the derivative of y = x2−1 . I We take the natural logarithm of both sides to get r x 2 + 1 ln y = ln 4 x 2 − 1 I Using the rules of logarithms to expand the R.H.S. we get 1 x 2 + 1 1 h i 1 1 ln y = ln = ln(x 2 +1)−ln(x 2 −1) = ln(x 2 +1)− ln(x 2 −1) 4 x 2 − 1 4 4 4 I Differentiating both sides with respect to x, we get 1 dy 1 2x 1 2x x x = · − · = − y dx 4 (x 2 + 1) 4 (x 2 − 1) 2(x 2 + 1) 2(x 2 − 1)

Annette Pilkington Natural Logarithm and Natural Exponential I Convertingy to a function of x, we get r " # dy x 2 + 1 x x = 4 − dx x 2 − 1 2(x 2 + 1) 2(x 2 − 1)

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Logarithmic differentiation q 4 x2+1 Example Find the derivative of y = x2−1 . I We take the natural logarithm of both sides to get r x 2 + 1 ln y = ln 4 x 2 − 1 I Using the rules of logarithms to expand the R.H.S. we get 1 x 2 + 1 1 h i 1 1 ln y = ln = ln(x 2 +1)−ln(x 2 −1) = ln(x 2 +1)− ln(x 2 −1) 4 x 2 − 1 4 4 4 I Differentiating both sides with respect to x, we get 1 dy 1 2x 1 2x x x = · − · = − y dx 4 (x 2 + 1) 4 (x 2 − 1) 2(x 2 + 1) 2(x 2 − 1)

I Multiplying both sides by y, we get dy h x x i = y − dx 2(x 2 + 1) 2(x 2 − 1)

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Logarithmic differentiation q 4 x2+1 Example Find the derivative of y = x2−1 . I We take the natural logarithm of both sides to get r x 2 + 1 ln y = ln 4 x 2 − 1 I Using the rules of logarithms to expand the R.H.S. we get 1 x 2 + 1 1 h i 1 1 ln y = ln = ln(x 2 +1)−ln(x 2 −1) = ln(x 2 +1)− ln(x 2 −1) 4 x 2 − 1 4 4 4 I Differentiating both sides with respect to x, we get 1 dy 1 2x 1 2x x x = · − · = − y dx 4 (x 2 + 1) 4 (x 2 − 1) 2(x 2 + 1) 2(x 2 − 1)

I Multiplying both sides by y, we get dy h x x i = y − dx 2(x 2 + 1) 2(x 2 − 1)

I Convertingy to a function of x, we get r " # dy x 2 + 1 x x = 4 − dx x 2 − 1 2(x 2 + 1) 2(x 2 − 1)

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Summary of formulas

ln(x)

a ln(ab) = ln a + ln b, ln( ) = ln a − ln b b ln ax = x ln a

lim ln x = ∞, lim ln x = −∞ x→∞ x→0

d 1 d g 0(x) ln |x| = , ln |g(x)| = dx x dx g(x) Z 1 dx = ln |x| + C x Z g 0(x) dx = ln |g(x)| + C. g(x)

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries Summary of methods

Logarithmic Differentiation (Finding formulas for inverse functions) Finding of inverse functions (using formula from lecture 1). Calculating Limits Calculating Derivatives Calculating (including definite integrals)

Annette Pilkington Natural Logarithm and Natural Exponential