Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x^In x
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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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05:36Change of Base Property
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
Textbook Question
At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.
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Textbook Question
{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?
f(x) = sin x on [−π,π]
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Textbook Question
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=tan x; (1,π/4)
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Textbook Question
Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.
h(x) = √x (√x-x³/²)
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Textbook Question
A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s (t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a.
s(t) = -16t2 + 128t + 192; a = 2
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