Textbook Question
Find y'' for the following functions.
y = tan x
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 3.30
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(s) = √s/4
Verified step by step guidance1
Step 1: Rewrite the function f(s) = \(\frac{\sqrt{s}\)}{4} in a form that is easier to differentiate. Recall that \(\sqrt{s}\) can be expressed as s^{1/2}. Therefore, f(s) = \(\frac{s^{1/2}\)}{4}.
Step 2: Apply the constant multiple rule of differentiation. The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function. Here, the constant is \(\frac{1}{4}\). So, we need to find the derivative of s^{1/2} and then multiply it by \(\frac{1}{4}\).
Step 3: Differentiate s^{1/2} using the power rule. The power rule states that if f(x) = x^n, then f'(x) = nx^{n-1}. Applying this to s^{1/2}, we get \(\frac{1}{2}\)s^{-1/2}.
Step 4: Multiply the result from Step 3 by the constant \(\frac{1}{4}\) from Step 2. This gives us \(\frac{1}{4}\) \(\times\) \(\frac{1}{2}\)s^{-1/2}.
Step 5: Simplify the expression obtained in Step 4. Combine the constants \(\frac{1}{4}\) and \(\frac{1}{2}\) to get \(\frac{1}{8}\). Therefore, the derivative of f(s) is \(\frac{1}{8}\)s^{-1/2}.
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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 at which a function changes at any given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. Derivatives are fundamental in calculus for understanding the behavior of functions, including their slopes and rates of change.
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Derivatives
Power Rule
The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions and can be applied to functions involving roots.
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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 g(x) that is composed with another function f(x), the derivative is found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is essential for differentiating complex functions, including those involving square roots.
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Related Practice
Textbook Question
Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.
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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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Explain why b^x = e^xlnb.