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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 7.3.126
Chapter 7, Problem 7.3.126

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.
126. eʸ = y^(ln x)

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Start with the given equation: \(e^{y} = y^{\ln x}\). Our goal is to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).
Take the natural logarithm of both sides to simplify the expression and make differentiation easier: \(\ln(e^{y}) = \ln(y^{\ln x})\).
Use logarithm properties to rewrite both sides: the left side becomes \(y\) (since \(\ln(e^{y}) = y\)), and the right side becomes \((\ln x) \cdot \ln y\) (since \(\ln(a^{b}) = b \ln a\)). So, we have \(y = (\ln x)(\ln y)\).
Differentiate both sides implicitly with respect to \(x\). Remember that \(y\) is a function of \(x\), so apply the chain rule when differentiating terms involving \(y\). For the left side, \(\frac{d}{dx}[y] = \frac{dy}{dx}\). For the right side, use the product rule: \(\frac{d}{dx}[(\ln x)(\ln y)] = \frac{1}{x} \ln y + (\ln x) \cdot \frac{1}{y} \frac{dy}{dx}\).
After differentiating, collect all terms involving \(\frac{dy}{dx}\) on one side and factor it out. Then solve for \(\frac{dy}{dx}\) algebraically to express the derivative explicitly in terms of \(x\) and \(y\).

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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. By taking the natural logarithm of both sides, the expression simplifies, allowing the use of implicit differentiation and product or chain rules more easily.
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Logarithmic Differentiation

Implicit Differentiation

Implicit differentiation is used when a function is defined implicitly rather than explicitly. It involves differentiating both sides of an equation with respect to the independent variable, treating dependent variables as functions, and then solving for the derivative.
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Finding The Implicit Derivative

Properties of Logarithms and Exponents

Understanding the properties of logarithms and exponents, such as ln(a^b) = b ln(a) and e^{ln(x)} = x, is essential for simplifying expressions during differentiation. These properties help rewrite complex expressions into forms that are easier to differentiate.
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Change of Base Property
Related Practice
Textbook Question

Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.

f(x)=(x+1)², x≥-1

Textbook Question

137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.

Textbook Question

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

y = (x^2 - 2x + 2)e^(x)

Textbook Question

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

25. y = ln(ln(x))

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Textbook Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.


7. lim (x → 2) (x - 2) / (x² - 4)

Textbook Question

Solve the differential equation in Exercises 9–22.

11. (dy/dx) = e^(x-y)

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