Merced College; Don Power

MATH 4B -- LAB -- DERIVATIVES OF INVERSE FUNCTIONS NAME ________________

Example 1: Find the derivative of . The derivative of its inverse function x² is known.

Procedure:
Identify the goal: Find y¢ if ...	y=
Solve for x	x=y^2
Differentiate implicitly	1=2y×y¢
Solve for y¢	y¢=1/2y and notice from earlier that y=
Write in terms of x	Dx()=1/(2)

Example 2: Find the derivative of e^x. The derivative of its inverse ln(x) is D_x ln(x)=1/x.

Procedure:
Identify the goal: Find y¢ if ...	y=e^x
Solve for x	x=ln(y)
Differentiate implicitly	1= y¢/y
Solve for y¢	y¢=y and notice from earlier that y=e^x
Write in terms of x	Dx(e^x)=e^x

Example 3: Find the derivative of sin^-1 x. The derivative of its inverse function sin(x) is known.

Procedure:
Identify the goal: Find y¢ if ...	y=sin^-1 x
Solve for x	x=sin(y)
Differentiate implicitly	1=cos(y)×y¢
Solve for y¢	y¢=1/cos(y) and notice from earlier that y=sin^-1 x
Write in terms of x	Dx(sin^-1 x)=1/cos(sin^-1 x)=

Calculations for the last line:
	We define q = sin^-1(x).
	Then sin(q) = x/1.
	Label the triangle accordingly, with q, x and 1.
	Calculate the third side and label it.
	Then cos(sin^-1x) = cos(q) =

Or, use a Pythagorean identity: cos(u) = and let u = sin^-1(x)

Example 4: Find the derivative of g(x) when the derivative of its inverse function f(x) is known.

Procedure:
Identify the goal: Find y¢ if ...	y=g(x)
Solve for x	x=f(y)
Differentiate implicitly	1=f¢(y)×y¢
Solve for y¢	y¢=1/f¢(y) and notice from earlier that y=g(x)
Write in terms of x

ASSIGNMENT: Use this procedure to derive the formula for the derivatives of:

1. sec^-1(x). Adapt the triangle procedure in example 3 for use with the secant.

2. sinh^-1(x). Use these facts: (a) The derivative of sinh(x) is cosh(x); (b) cosh²(y)-sinh²(y) = 1

(In your answer, you will have to rewrite cosh(y) as and then as before you can get an expression entirely in x)

Return to: Merced College; Don Power Updated 7/10/06 by Don Power