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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.AAE.9
Chapter 7, Problem 7.AAE.9

In Exercises 9 and 10, use implicit differentiation to find dy/dx.
9. y^e^x = x^y + 1

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Start with the given equation: \(y^{e^{x}} = x^{y} + 1\).
Differentiate both sides of the equation with respect to \(x\) using implicit differentiation. Remember that \(y\) is a function of \(x\), so when differentiating terms involving \(y\), apply the chain rule.
For the left side, \(y^{e^{x}}\), use logarithmic differentiation: take the natural logarithm of both sides or apply the formula for differentiating \(u^{v}\) where both \(u\) and \(v\) depend on \(x\). The derivative is \(\frac{d}{dx} y^{e^{x}} = y^{e^{x}} \left( e^{x} \ln(y) + e^{x} \frac{1}{y} \frac{dy}{dx} \right)\).
For the right side, differentiate \(x^{y}\) using implicit differentiation: \(\frac{d}{dx} x^{y} = x^{y} \left( \frac{y}{x} + \ln(x) \frac{dy}{dx} \right)\), and the derivative of the constant \(1\) is \(0\).
After differentiating both sides, collect all terms involving \(\frac{dy}{dx}\) on one side and factor it out. Then solve for \(\frac{dy}{dx}\) to express it explicitly in terms of \(x\), \(y\), and known functions.

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Key Concepts

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

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when y is defined implicitly rather than explicitly in terms of x. It involves differentiating both sides of the equation with respect to x, treating y as a function of x, and applying the chain rule when differentiating terms involving y.
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Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. In implicit differentiation, it allows us to differentiate terms involving y by multiplying the derivative of the outer function by dy/dx, since y depends on x.
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Intro to the Chain Rule

Logarithmic Differentiation

Logarithmic differentiation is useful when differentiating expressions where variables appear as both bases and exponents, such as y^e^x or x^y. Taking the natural logarithm of both sides simplifies the differentiation process by converting products and powers into sums and products, making implicit differentiation more manageable.
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Logarithmic Differentiation
Related Practice
Textbook Question

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

Textbook Question

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

Textbook Question

15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.

Textbook Question

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.

122. y = (ln x)^(ln x)

Textbook Question

Find the limits in Exercises 1–6.

5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))

Textbook Question

7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.

a. lim(x→∞)A(t)

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