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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.9.79
Chapter 3, Problem 3.9.79

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x^In x

Verified step by step guidance
1
Step 1: Start by taking the natural logarithm of both sides of the equation y = x^(ln x). This gives us ln(y) = ln(x^(ln x)).
Step 2: Use the property of logarithms that allows you to bring the exponent down: ln(y) = ln x * ln x.
Step 3: Differentiate both sides with respect to x. For the left side, use implicit differentiation: (1/y) * dy/dx. For the right side, use the product rule: d(ln x * ln x)/dx = ln x * (1/x) + ln x * (1/x).
Step 4: Solve for dy/dx by multiplying both sides by y: dy/dx = y * (ln x/x + ln x/x).
Step 5: Substitute back y = x^(ln x) to express dy/dx in terms of x: dy/dx = x^(ln x) * (2 ln x/x).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables raised to variable powers. By taking the natural logarithm of both sides of the function, we can simplify the differentiation process, especially when dealing with complex expressions. This method is particularly useful for functions like f(x) = x^ln(x), where both the base and the exponent are variable.
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Logarithmic Differentiation

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential when applying logarithmic differentiation, as it allows us to handle the derivatives of the inner and outer functions effectively.
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Intro to the Chain Rule

Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses of each other, with exponential functions having the form y = a^x and logarithmic functions expressed as x = log_a(y). Understanding their properties is crucial for logarithmic differentiation, as it involves manipulating these functions to simplify the differentiation process. For instance, knowing that ln(a^b) = b*ln(a) helps in breaking down the function f(x) = x^ln(x) into manageable parts.
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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

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

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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

13–40. Evaluate the derivative of the following functions.

f(x) = 1/tan^−1(x²+4)

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