3.1 Derivatives of Polynomials and Exponential Functions 3.2 The

CH 3: Diﬀerentiation Rules

3.1 Derivatives of Polynomials and Exponential Functions

d 1. f(x) is another notation for f 0(x) that points out the variable you must take the derivative dx with respect to. d 2. constants: (constant) = 0 dx d d 3. the power rule: (xn) = nxn−1, for all real numbers n, particularly (x) = 1 dx dx 4. note that the power rule is a shortcut in ﬁnding the slope of the tangent line of the curve xn d d d d 5. the sum rule: f(x) + g(x) = (f(x) + g(x)). Example: (x7 + x) = 7x6 + 1 dx dx dx dx

d d d d √ 1 − 1 6. the diﬀerence rule: f(x) − g(x) = (f(x) − g(x)). Example: ( x − 3) = x 2 − 0 = dx dx dx dx 2 1 √ 2 x d d d 7. the constant multiple rule: (c · f(x)) = c f(x). Example: (3x−7) = 3 · (−7)x−6 = dx dx dx −21x−6

8. recall e! The special property that e has is that the function ex has a tangent line with a slope eh − e0 of 1 at the point x = 0, i.e. if f(x) = ex, then f 0(0) = lim = 1 h→0 h d 9. the derivative of the natural exponential function: (ex) = ex, dx 10. recall: the secant line to a curve is the line that goes through 2 points on the graph (if the graph represents distance, then its slope is the average speed between the two points)

11. recall: the tangent line to a curve is the line that touches the graph exactly once at that point (if the graph represents distance, then its slope is the instantaneous speed at that point)

12. we deﬁne the normal line to a curve is the line that is perpendicular to the tangent line (if we have a 3-dimensional surface, then we have a tangent plane that touches the surface at exactly one point, and the normal line is perpendicular to each line in this tangent plane and it helps ﬁnd the equation of the plane – you’ll see this in MA1115)

3.2 The product and the quotient rules

d d d 1. ﬁrst note that (f(x) · g(x)) 6= f(x) · g(x) (i.e. (f · g)0 6= f 0 · g0). Check this on f(x) = x3 dx dx dx and g(x) = x10 d d d 2. the product rule: (f(x) · g(x)) = f(x) g(x) + g(x) f(x) (i.e. (f · g)0 = f · g0 + f 0 · g). dx dx dx

1 d d f(x) f(x) 0 3. similarly note that 6= dx (i.e. ( f )0 6= f ). Check this on f(x) = x3 and g(x) = x10 dx g(x) d g g0 dx g(x) d d d f(x) g(x) f(x) − f(x) g(x) 0 0 4. the quotient rule: = dx dx (i.e. ( f )0 = f ·g−f·g ) dx g(x) g2(x) g g2

3.3 Derivatives of the Trig functions

1. (sin x)0 = cos x (sec x)0 = sec x · tan x

2. (cos x)0 = − sin x (csc x)0 = − csc x · cot x

3. (tan x)0 = sec2 x (cot x)0 = − csc2 x sin x 4. lim = 1 x→0 x

3.4 The Chain Rule

1. The chain rule: Let f and g be two functions. If g differentiable at x, and f is differentiable at g(x), then the composition F = f ◦ g is defined and its derivative is F 0(x) = f 0 g(x) · g0(x).

2. (ax)0 = ax ln a, for a > 0