Textbook Question
If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.
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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 3.9.65
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
y = (cos x) In cos²x
Verified step by step guidance1
First, identify the function y = (cos x) ln(cos²x). Notice that it is a product of two functions: u(x) = cos(x) and v(x) = ln(cos²x).
Apply the product rule for derivatives, which states that if y = u(x) * v(x), then y' = u'(x) * v(x) + u(x) * v'(x).
Calculate the derivative of u(x) = cos(x). The derivative u'(x) is -sin(x).
Simplify v(x) = ln(cos²x) using the properties of logarithms: ln(cos²x) = 2 ln(cos(x)). Now, find the derivative v'(x). The derivative of ln(cos(x)) is -tan(x), so v'(x) = 2(-tan(x)) = -2tan(x).
Substitute u'(x), v(x), u(x), and v'(x) into the product rule formula: y' = (-sin(x)) * ln(cos²x) + cos(x) * (-2tan(x)). Simplify the expression to find the derivative.
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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 can be calculated using various rules, such as the product rule, quotient rule, and chain rule.
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Derivatives
Logarithmic Properties
Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). These properties are particularly useful in calculus for simplifying complex functions before differentiation, making it easier to compute derivatives.
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Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential when dealing with functions that are nested within each other, allowing for the correct application of differentiation.
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Related Practice
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Textbook Question
Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?
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Textbook Question
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
y = 4 log₃(x²−1)
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