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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 7.1.20
Chapter 7, Problem 7.1.20

7–28. Derivatives Evaluate the following derivatives.


d/dt (t^{1/t})

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Step 1: Recognize that the function t^(1/t) involves both a variable base and a variable exponent. To differentiate it, use logarithmic differentiation. Start by taking the natural logarithm of both sides: let y = t^(1/t), then ln(y) = (1/t) * ln(t).
Step 2: Differentiate both sides of the equation with respect to t. For the left-hand side, use implicit differentiation: d/dt[ln(y)] = (1/y) * dy/dt. For the right-hand side, apply the product rule to differentiate (1/t) * ln(t).
Step 3: Apply the product rule to (1/t) * ln(t). The product rule states that d/dt[u*v] = u'v + uv'. Here, u = 1/t and v = ln(t). Differentiate u and v separately: u' = -1/t^2 and v' = 1/t.
Step 4: Substitute the derivatives into the product rule formula: d/dt[(1/t) * ln(t)] = (-1/t^2) * ln(t) + (1/t) * (1/t). Simplify the expression to combine terms.
Step 5: Solve for dy/dt by multiplying through by y (recall y = t^(1/t)) to isolate dy/dt. Substitute back y = t^(1/t) into the final expression to express the derivative in terms of t.

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Key Concepts

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

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that allows us to understand how a function behaves locally. The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point, indicating how the function's output changes as the input changes.
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Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions that involve exponentiation or logarithms, as seen in the given expression.
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Exponential Functions

Exponential functions are functions of the form f(t) = a^g(t), where a is a constant and g(t) is a function of t. In the context of the problem, t^{1/t} can be rewritten using properties of exponents, which can simplify the differentiation process. Understanding how to manipulate and differentiate exponential functions is crucial for solving problems involving derivatives of such forms.
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Related Practice
Textbook Question

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Textbook Question

7–28. Derivatives Evaluate the following derivatives.


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11–15. Identities Prove each identity using the definitions of the hyperbolic functions.


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Harmonic sum In Chapter 10, we will encounter the harmonic sum 1 + 1/2 + 1/3 + ⋯ + 1/n. Use a left Riemann sum to approximate ∫[1 to n+1] (dx/x) (with unit spacing between the grid points) to show that 1 + 1/2 + 1/3 + ⋯ + 1/n > ln(n + 1). Use this fact to conclude that lim (n → ∞) (1 + 1/2 + 1/3 + ⋯ + 1/n) does not exist.