Derivatives of Basic Functions Power Rule: Trigonometric Functions: d [xn] = nxn−1; where n is any real number dx d d [sin(x)] = cos(x) [cos(x)] = − sin(x) dx dx We can use these two to find the derivatives of the re- Derivative of a Constant: maining trig functions: d [c] = 0 d d dx [tan(x)] = sec2(x) [cot(x)] = − csc2(x) dx dx d d [sec(x)] = sec(x) tan(x) [csc(x)] = − csc(x) cot(x) dx dx Exponential Functions: With base a: Inverse Trigonometric Functions: d [ax] = ln(a) · ax dx d d 1 [arctan(x)] = tan−1(x) = With base e, this becomes: dx dx 1 + x2 d d d −1 1 [ex] = ex [arcsin(x)] = sin (x) = p ; −1 < x < 1 dx dx dx 1 − x2 d d 1 [arccos(x)] = cos−1(x) = −p ; −1 < x < 1 dx dx 1 − x2 Logarithmic Functions: With base a: d 1 1 [log (x)] = · dx a ln(a) x With base e, this becomes: d 1 [ln(x)] = dx x Differentiation Rules and Techniques Constant Multiple Rule: d [c · f(x)] = c · f 0(x) dx Sum/Difference Rule: d [f(x) ± g(x)] = f 0(x) ± g0(x) dx Product Rule: d [f(x) · g(x)] = f 0(x) · g(x) + f(x) · g0(x) dx d 0 0 Remember, dx [f(x) · g(x)] 6= f (x) · g (x), i.e., you cannot simply take the derivatives of each function and multiply them together. For products of 3 or more functions, there is a similar pattern: d [f(x) · g(x) · h(x)] = f 0(x) · g(x) · h(x) + f(x) · g0(x) · h(x) + f(x) · g(x) · h0(x) dx Quotient Rule: d f(x) f 0(x) · g(x) − f(x) · g0(x) = dx g(x) [g(x)]2 d h f(x) i f 0(x) Remember, dx g(x) 6= g0(x) , i.e., you cannot simply take the derivatives of each function and divide them. Chain Rule: d d [(f ◦ g)(x)] = [f(g(x))] = f 0(g(x)) · g0(x) = (derivative of the outside) · (derivative of the inside) dx dx Tangents to Parametric Curves: Given a parametric curve x = f(t); y = g(t) then: dy dy dt dx = ; if 6= 0. dx dx dt dt Implicit Differentiation: Used on equations that define a function implicitly (as opposed to explicitly as y = f(x)). 1. Differentiate both sides of the equations w.r.t x, treating y as a function of x and using the chain rule as needed. dy 2. Solve for dx : dy (a) Collect all terms involving dx in the LHS (move all other terms to RHS), dy (b) then factor dx out of the LHS, dy (c) and finally divide through by LHS factor that does not involve dx . Logarithmic Differentiation: The book is using the phrase \logarithmic differentiation" to refer to two different things in this section: The first, and what most people mean when they say \logarithmic differentiation", is a technique that can be used when differentiating a more complicated function y = f(x). Here are the steps: 1. Take the natural log of both sides: ln(y) = ln (f(x)) (we could have used any base log, but ln is a little nicer to differentiate). 2. Use properties of logs to expand the RHS as much as possible 3. Differentiate (using implicit differentiation on the LHS). dy 4. Solve for dx . dy 5. Substitute f(x) in for y (want final answer for dx in terms of x only!). The other type of problem that comes up in this section is the situation where you want to differentiate a function that looks like y = loga(something rather complicated and unpleasent (typically the \loga" will be \ln"). In this case, you simply use properties of logarithms to expand the RHS as much as possible, and then differentiate. dy So, there is no implicit differentiation on the LHS, solving for dx , or substituting in or multiplying by y = f(x) at the end..