Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
h(t) = t²/2 + 1
2
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Ch. 3 - Derivatives
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
Chapter 3, Problem 3.9.32
Find the derivative of the following functions.
y = In(e^x + e^-x)
Verified step by step guidance1
Step 1: Recognize that the function y = ln(e^x + e^-x) is a composition of functions, where the outer function is the natural logarithm ln(u) and the inner function is u = e^x + e^-x.
Step 2: Apply the chain rule for differentiation, which states that if y = ln(u), then the derivative dy/dx = (1/u) * (du/dx).
Step 3: Differentiate the inner function u = e^x + e^-x with respect to x. The derivative of e^x is e^x, and the derivative of e^-x is -e^-x.
Step 4: Substitute the derivative of the inner function back into the chain rule formula. This gives dy/dx = (1/(e^x + e^-x)) * (e^x - e^-x).
Step 5: Simplify the expression if possible. The derivative of y with respect to x is dy/dx = (e^x - e^-x) / (e^x + e^-x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) or dy/dx, and it can be calculated using various rules such as the power rule, product rule, and chain rule.
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Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is the inverse function of the exponential function e^x. Understanding the properties of natural logarithms, such as ln(a*b) = ln(a) + ln(b) and ln(a^b) = b*ln(a), is essential for differentiating functions that involve logarithmic expressions.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where a and b are constants, and e is the base of the natural logarithm. These functions exhibit rapid growth or decay and are characterized by their unique property that the derivative of e^x is e^x. In the context of the given function, recognizing how to differentiate expressions involving e^x and e^(-x) is crucial for finding the derivative.
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Related Practice
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Exercise 48