Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = x² cosh² 3x
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.16
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.16Chapter 7, Problem 7.1.16
7–28. Derivatives Evaluate the following derivatives.
d/dt ((sin t)^{√t})
Verified step by step guidance1
Step 1: Recognize that the given function is of the form \( f(t) = (\sin t)^{\sqrt{t}} \), which is a composite function involving both a power and a trigonometric function. To differentiate it, we will use the logarithmic differentiation technique.
Step 2: Take the natural logarithm of both sides to simplify the differentiation process. Let \( y = (\sin t)^{\sqrt{t}} \), then \( \ln y = \sqrt{t} \cdot \ln(\sin t) \).
Step 3: Differentiate both sides with respect to \( t \). Use the chain rule on the left-hand side and the product rule on the right-hand side. The derivative of \( \ln y \) is \( \frac{1}{y} \cdot \frac{dy}{dt} \), and the derivative of \( \sqrt{t} \cdot \ln(\sin t) \) requires applying the product rule.
Step 4: Apply the product rule to \( \sqrt{t} \cdot \ln(\sin t) \). The derivative of \( \sqrt{t} \) is \( \frac{1}{2\sqrt{t}} \), and the derivative of \( \ln(\sin t) \) is \( \frac{1}{\sin t} \cdot \cos t \) (using the chain rule). Combine these results.
Step 5: Solve for \( \frac{dy}{dt} \) by multiplying through by \( y \), which is \( (\sin t)^{\sqrt{t}} \). Substitute \( y \) back into the expression to obtain 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.
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 two functions u and v, 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 differentiating functions where one function is nested within another.
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05:02
Intro to the Chain Rule
Product Rule
The Product Rule is a method for differentiating products of two functions. It states that if you have two functions u(t) and v(t), the derivative of their product is given by u'v + uv'. This rule is particularly useful when dealing with expressions where two functions are multiplied together, such as in the case of (sin t)^{√t}.
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05:18The Product Rule
Exponential Functions and Logarithmic Differentiation
Exponential functions, especially those involving variables in the exponent, can be complex to differentiate directly. Logarithmic differentiation is a technique that simplifies this process by taking the natural logarithm of both sides of the equation. This allows for easier manipulation and differentiation, particularly when dealing with functions like (sin t)^{√t}, where the exponent is itself a function of t.
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06:30Logarithmic Differentiation
Related Practice
Textbook Question
Logarithm properties Use the integral definition of the natural logarithm to prove that ln(x/y) = ln x - ln y.
Textbook Question
What is the domain of sech⁻¹ x? How is sech⁻¹ x defined in terms of the inverse hyperbolic cosine?
Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = tanh²x
Textbook Question
7–28. Derivatives Evaluate the following derivatives.
d/dx (sin (ln x))
Textbook Question
63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.
limₕ→₀ (1 + 2h)^{1/h}