3.5 and 3.6 – Implicit and Inverse Functions Math 1304 Calculus I 3.5 and 3.6 – Implicit and Inverse Functions

Implicit and Explicit Functions Explicit: y = f(x) Implicit: F(x,y)=0 Example: implicit explicit

Implicit Differentiation If f(x) = g(x), then f’(x) = g’(x) Example: x2 + y2 = 1

Inverse f and g are inverse if: y = f(x) iff x = g(y) Also f and g are inverse if f(g(y) = y and g(f(x) = x Examples Exponential and Log y = ln(x) iff x = ey Trigonometric: sin and arcsin y = arcsin(x) iff x = sin(y)

Derivatives of inverse functions Proof? (in class)

Derivative of Logarithms If F(x) = loga(f(x)), then F’(x) = (1/ln a) f’(x)/f(x) Proof? (in class) Special case: If F(x) = ln(f(x)), then F’(x) = f’(x)/f(x)

A new good working set of rules Constants: If F(x) = c, then F’(x) = 0 Powers: If F(x) = f(x)n, then F’(x) = n f(x)n-1 f’(x), where n is real Exponentials: If F(x) = af(x), then F’(x) = (ln a) af(x) f’(x) Logarithms: If F(x) = loga(f(x)), then F’(x) = (1/ln a) f’(x)/f(x) Trigonometric functions: If F(x) = sin(f(x)), then F’(x) = cos(f(x)) f’(x) If F(x) = csc(f(x)), then F’(x) = - csc(f(x)) cot(f(x)) f’(x) If F(x) = cos(f(x)), then F’(x) = - sin(f(x)) f’(x) If F(x) = sec(f(x)), then F’(x) = sec(f(x)) tan(f(x)) f’(x) If F(x) = tan(f(x)), then F’(x) = sec2(f(x)) f’(x) If F(x) = cot(f(x)), then F’(x) = - csc2(f(x)) f’(x) Inverse trig functions: If f(x) = arcsin(x), then f’(x) = 1/√ (1-x2) If f(x) = arccos(x), then f’(x) = -1/√ (1-x2) If f(x) = arctan(x), then f’(x) = 1/(1+x2) Scalar multiplication: If F(x) = c f(x), then F’(x) = c f’(x) Sum: If F(x) = g(x) + h(x), then F’(x) = g’(x) + h’(x) Difference: If F(x) = g(x) - h(x), then F’(x) = g’(x) - h’(x) Multiple sums: derivative of sum is sum of derivatives Linear combinations: derivative of linear combination is linear combination of derivatives Product: If F(x) = g(x) h(x), then F’(x) = g’(x) h(x) + g(x)h’(x) Multiple products: If F(x) = g(x) h(x) k(x), then F’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x) Quotient: If F(x) = g(x)/h(x), then F’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2 Composition: If F = fog is a composite, defined by F(x) = f(g(x)) then F'(x) = f'(g(x))g'(x)

Logarithmic Differentiation Sometimes it helps to take the ln of both sides of an equation before differentiation. Then solve for y’ Examples: y = f(x)g(x)

Use of logarithmic differentiation Prove general power law Quick proof of product rule