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:

2

22

1

1

xy

yx

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 Logy = ln(x) iff x = ey

Trigonometric: sin and arcsiny = arcsin(x) iff x = sin(y)

Derivatives of inverse functions

2

2

2

1

1)arctan(

1

1)arccos(

1

1)arcsin(

xx

dx

dx

xdx

dx

xdx

d

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