Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.1.51

Chapter 7, Problem 7.1.51

Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).

Verified step by step guidance

First, identify the function given: \(f(x) = x^{3} - 3x^{2} - 1\) with the domain \(x \geq 2\). We are asked to find the derivative of the inverse function \(f^{-1}(x)\) at the point where \(x = -1\), which corresponds to \(f(3) = -1\).

Recall the formula for the derivative of the inverse function: \(\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}\). This means we need to find \(f'(x)\) and evaluate it at \(x = f^{-1}(-1)\), which is \(x=3\) since \(f(3) = -1\).

Compute the derivative of \(f(x)\): \(f'(x) = \frac{d}{dx} (x^{3} - 3x^{2} - 1) = 3x^{2} - 6x\).

Evaluate \(f'(x)\) at \(x=3\): substitute \(3\) into the derivative to get \(f'(3) = 3(3)^{2} - 6(3)\).

Finally, use the inverse derivative formula to find \(\frac{d}{dx} f^{-1}(-1) = \frac{1}{f'(3)}\). This gives the derivative of the inverse function at the point \(x = -1\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Function and Its Derivative

The inverse function f⁻¹ reverses the effect of f, so f(f⁻¹(x)) = x. The derivative of the inverse at a point x is given by (df⁻¹/dx)(x) = 1 / (df/dx)(f⁻¹(x)), provided f is differentiable and its derivative is nonzero at that point.

Chain Rule

The chain rule helps differentiate composite functions. For inverse functions, it relates the derivatives of f and f⁻¹ by differentiating the identity f(f⁻¹(x)) = x, leading to the formula for the derivative of the inverse function.

Evaluating Derivatives at Specific Points

To find the derivative of the inverse at a given x, identify the corresponding point y = f⁻¹(x). Then compute the derivative of f at y and use it to find (df⁻¹/dx)(x). This requires careful substitution and evaluation of the original function and its derivative.

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Related Practice

Textbook Question

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = (x + b) / (x − 2), b > −2 and constant

Textbook Question

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

73. y = log₄ x + log₄ x²

Textbook Question

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = cos(e^(-θ^2))

Textbook Question

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

79. y = θ sin(log₇ θ)

Textbook Question

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

14. y = ln(2θ+2)

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Textbook Question

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

31. y=arccot(√t)

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