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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.3.115
Chapter 7, Problem 7.3.115

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
115. y = (x + 1)ˣ

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Recognize that the function is of the form \(y = (x + 1)^x\), where both the base and the exponent depend on \(x\). This makes logarithmic differentiation a suitable method.
Take the natural logarithm of both sides to simplify the expression: \(\ln y = \ln \left( (x + 1)^x \right)\).
Use the logarithm power rule to bring the exponent down: \(\ln y = x \cdot \ln (x + 1)\).
Differentiate both sides with respect to \(x\). Remember to use implicit differentiation on the left side and the product rule on the right side: \(\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( x \ln (x + 1) \right)\).
Apply the product rule to the right side: \(\frac{d}{dx} \left( x \ln (x + 1) \right) = \ln (x + 1) + x \cdot \frac{1}{x + 1}\), then multiply both sides by \(y\) to solve for \(\frac{dy}{dx}\).

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Key Concepts

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

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions where the variable appears both in the base and the exponent, such as y = (x + 1)^x. By taking the natural logarithm of both sides, the expression simplifies, allowing the use of implicit differentiation to find dy/dx more easily.
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Logarithmic Differentiation

Implicit Differentiation

Implicit differentiation involves differentiating both sides of an equation with respect to x when y is defined implicitly. After taking the logarithm, y is expressed in terms of ln(y), so differentiating requires applying the chain rule to ln(y), treating y as a function of x.
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Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. In logarithmic differentiation, it is applied to differentiate ln(y), resulting in (1/y) * dy/dx, which helps isolate dy/dx and find the derivative of the original function.
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Intro to the Chain Rule
Related Practice
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