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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.26
Chapter 7, Problem 7.1.26

Evaluate the following derivatives.


d/dx (x^{x¹⁰})

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Step 1: Recognize that the function x^(x^10) is a composite function involving both a power and an exponential term. To differentiate it, use the logarithmic differentiation technique.
Step 2: Let y = x^(x^10). Take the natural logarithm of both sides: ln(y) = ln(x^(x^10)). Simplify using logarithmic properties: ln(y) = x^10 * ln(x).
Step 3: Differentiate both sides with respect to x. For the left-hand side, use implicit differentiation: (1/y) * dy/dx. For the right-hand side, apply the product rule to differentiate x^10 * ln(x).
Step 4: The derivative of x^10 * ln(x) is (x^10)' * ln(x) + x^10 * (ln(x))'. Compute these derivatives: (x^10)' = 10x^9 and (ln(x))' = 1/x.
Step 5: Substitute the derivatives back into the equation and solve for dy/dx. Multiply through by y (which is x^(x^10)) to express the derivative in terms of the original function.

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Key Concepts

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

Chain Rule

The Chain Rule is a fundamental differentiation technique used to find the derivative of composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for evaluating derivatives of functions where one function is nested within another.
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Product Rule

The Product Rule is a method for differentiating products of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is particularly useful when dealing with functions that are multiplied together, such as in the case of x raised to a power that is also a function of x.
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Exponential Functions and Logarithmic Differentiation

Exponential functions, particularly those of the form f(x) = x^g(x), can be differentiated using logarithmic differentiation. This technique involves taking the natural logarithm of both sides, which simplifies the differentiation process, especially when the exponent is a function of x. By applying the properties of logarithms, one can transform the expression into a more manageable form for differentiation.
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Related Practice
Textbook Question

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫₁² (1 + ln x) x^x dx

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Textbook Question

Atmospheric pressure The pressure of Earth’s atmosphere at sea level is approximately 1000 millibars and decreases exponentially with elevation. At an elevation of 30,000 ft (approximately the altitude of Mt. Everest), the pressure is one-third the sea-level pressure. At what elevation is the pressure half the sea-level pressure? At what elevation is it 1% of the sea-level pressure?

Textbook Question

7–28. Derivatives Evaluate the following derivatives.


d/dx (x^{π})

Textbook Question

Bounds on e Use a left Riemann sum with at least n = 2 subintervals of equal length to approximate ln 2 = ∫[1 to 2] (dt/t) and show that ln 2 < 1. Use a right Riemann sum with n = 7 subintervals of equal length to approximate ln 3 = ∫[1 to 3] (dt/t) and show that ln 3 > 1.

Textbook Question

Evaluate ∫ 4ˣ dx.

Textbook Question

On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?