Textbook Question
Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.
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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 3.5.62
Find y'' for the following functions.
y = tan x
Verified step by step guidance1
Start by identifying the function y = tan(x). To find the second derivative y'', we first need to find the first derivative y'.
Recall the derivative of tan(x) with respect to x is sec^2(x). Therefore, y' = sec^2(x).
Next, we need to find the derivative of y' to obtain y''. This means we need to differentiate sec^2(x) with respect to x.
Use the chain rule to differentiate sec^2(x). Let u = sec(x), then y' = u^2. The derivative of u^2 with respect to x is 2u * du/dx.
Now, find du/dx where u = sec(x). The derivative of sec(x) is sec(x)tan(x). Substitute back to get y'' = 2sec(x) * sec(x)tan(x) = 2sec^3(x)tan(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Second Derivative
The second derivative of a function measures the rate of change of the first derivative. It provides information about the concavity of the function and can indicate points of inflection where the function changes from concave up to concave down or vice versa. In this context, finding y'' involves differentiating the function y = tan x twice.
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The Second Derivative Test: Finding Local Extrema
Differentiation Rules
Differentiation rules are formulas and techniques used to compute the derivative of functions. Key rules include the power rule, product rule, quotient rule, and chain rule. For the function y = tan x, the derivative can be found using the quotient rule, as tan x can be expressed as sin x / cos x.
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Trigonometric Derivatives
Trigonometric derivatives are specific derivatives of trigonometric functions, which have unique rates of change. For example, the derivative of tan x is sec² x. Understanding these derivatives is essential for solving problems involving trigonometric functions, such as finding the second derivative of y = tan x.
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Related Practice
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.
y= √x; P(4, 2)
Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(s) = √s/4
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Textbook Question
Find y'' for the following functions.
y = x sin x
Textbook Question
Use Theorem 3.10 to evaluate the following limits.
lim x🠂2 (sin (x-2)) / (x2 - 4)
Textbook Question
Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
y = 12s³-8s²+12s/4s
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