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AP Calculus AB: 6.4.1 Derivatives of Inverse Function

Mathematics12 CardsCreated 3 months ago

This section explains how to compute the derivative of an inverse function at a specific point using a known formula, without explicitly finding the inverse.

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Derivatives of Inverse Functions

You can calculate the derivative of an inverse function at a point without determining the actual inverse function.

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Key Terms

Term
Definition

Derivatives of Inverse Functions

You can calculate the derivative of an inverse function at a point without determining the actual inverse function.

note

  • The inverse of a function retains many of the properties of the original function.

  • To derive the formula for the derivative ...

If f is an invertible function, which of the following is not true?

If f is increasing then f −1 is decreasing.

If f (x) = (e^ x − e^ −x ) / 2 and f −1 (0) = 0, find the derivative of f −1 at x = 0.

1

If f (x) = x + ln x, where x > 0, and
f −1 (1 + e) = e, find the derivative of
f −1 at x = 1 + e.

e/e+1

Let f be a function. If f′(x)≥2, for any x,which of the following is true?

d/dx[f−1(x)]≤1/2, for any x

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AP Calculus AB: 6.4.1 Derivatives of Inverse Function
TermDefinition

Derivatives of Inverse Functions

You can calculate the derivative of an inverse function at a point without determining the actual inverse function.

note

  • The inverse of a function retains many of the properties of the original function.

  • To derive the formula for the derivative of an inverse, start with a relationship you know.

  • The composition of a function and its inverse is equal to x. You need to use implicit differentiation. Use the chain rule to differentiate both sides of that relationship.

  • Isolate the derivative of the inverse by dividing.

  • If you know the value of the inverse at a point, you can find the derivative of the inverse at that point.

  • In this example, you know the function and the value of the inverse at π. Your mission is to find the value of the
    derivative of the inverse at π.

  • Use the formula that you learned above. The derivative of f(x) = 2x + cos x is f ́(x) = 2 – sin x. Sine takes on values between –1 and 1, so the derivative lies between 1 and 3. It’s always positive, which means the function is increasing. Remember that increasing functions are invertible.

  • Once you have found the derivative of the original function and verified that the function is invertible, all you have to do is plug into the formula.

  • You have evaluated the derivative of the inverse of a function at a point, without determining the inverse itself!

If f is an invertible function, which of the following is not true?

If f is increasing then f −1 is decreasing.

If f (x) = (e^ x − e^ −x ) / 2 and f −1 (0) = 0, find the derivative of f −1 at x = 0.

1

If f (x) = x + ln x, where x > 0, and
f −1 (1 + e) = e, find the derivative of
f −1 at x = 1 + e.

e/e+1

Let f be a function. If f′(x)≥2, for any x,which of the following is true?

d/dx[f−1(x)]≤1/2, for any x

If f (x) = x ^3 + 3x, and f −1 (4) = 1, find the derivative of f −1 at x = 4.

1/6

If f (x) = sin x + e^ x + x and f −1 (1) = 0, find the derivative of f −1 at x = 1.

1/3

If f (x) = sin^2 x − 2x, and f −1 (0) = 0, find the derivative of f −1 at x = 0.

-1/2

If f (x) = x ^101 + 101x, and f −1 (102) = 1, find the derivative of f −1 at x = 102.

1/202

If f(x)=sinx−3x and f−1(−3π)=π, find the derivative of f−1 at x=−3π.

-1/4

If f(x)=x+e^x and f−1(1)=0, find the derivative of f−1 at x=1.

1/2