Textbook Question
Calculate the derivative of the following functions.
y = (e^x / x+1)⁸
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 66
Calculate the derivative of the following functions.
y = tan(xe^x)
Verified step by step guidance1
Step 1: Identify the outer function and the inner function. Here, the outer function is \( \tan(u) \) and the inner function is \( u = xe^x \).
Step 2: Apply the chain rule for differentiation, which states that \( \frac{dy}{dx} = \frac{d}{du}[\tan(u)] \cdot \frac{du}{dx} \).
Step 3: Differentiate the outer function \( \tan(u) \) with respect to \( u \). The derivative of \( \tan(u) \) is \( \sec^2(u) \).
Step 4: Differentiate the inner function \( u = xe^x \) with respect to \( x \). Use the product rule, which states that \( \frac{d}{dx}[uv] = u'v + uv' \), where \( u = x \) and \( v = e^x \).
Step 5: Combine the results from Step 3 and Step 4 to find \( \frac{dy}{dx} = \sec^2(xe^x) \cdot (e^x + xe^x) \).
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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 changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as in the case of y = tan(xe^x).
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Chain Rule
The chain rule is a technique for differentiating composite functions. If a function y is defined as a composition of two functions, such as y = f(g(x)), the chain rule states that the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is particularly useful when dealing with functions like tan(u), where u itself is a function of x, such as xe^x in this case.
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Related Practice
Textbook Question
Population growth Consider the following population functions.
b. What is the instantaneous growth rate at t=5?
p(t) = 600 (t²+3/t²+9)
Textbook Question
Population growth Consider the following population functions.
a. Find the instantaneous growth rate of the population, for t≥0.
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Textbook Question
A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
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Textbook Question
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c. Estimate the time when the instantaneous growth rate is greatest.
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Textbook Question
Let f(x) = x2 - 6x + 5.
Find the values of x for which the slope of the curve y = f(x) is 2.
1
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