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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.R.7
Chapter 3, Problem 3.R.7

Use differentiation to verify each equation.


d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).

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1
Start by differentiating the first term, x⁴. The derivative of x⁴ with respect to x is 4x³. This is because the power rule states that d/dx(xⁿ) = nxⁿ⁻¹.
Next, differentiate the second term, ln(x⁴ + 1). Use the chain rule here: the derivative of ln(u) is 1/u times the derivative of u. Let u = x⁴ + 1, so the derivative of u with respect to x is 4x³.
Apply the chain rule: the derivative of ln(x⁴ + 1) is (1/(x⁴ + 1)) * 4x³.
Combine the derivatives: the derivative of the entire expression x⁴ − ln(x⁴ + 1) is 4x³ - (4x³/(x⁴ + 1)).
Simplify the expression: factor out 4x³ from the terms to get 4x³(1 - 1/(x⁴ + 1)). Simplify further to obtain 4x⁷/(1 + x⁴), which matches the given equation.

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Key Concepts

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

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable, providing insights into the function's behavior, such as its slope at any given point. Techniques for differentiation include the power rule, product rule, quotient rule, and chain rule, each applicable in different scenarios.
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Chain Rule

The chain rule is a specific differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. This rule is essential for differentiating functions where one function is nested within another, allowing for the correct application of derivatives in complex expressions.
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Intro to the Chain Rule

Logarithmic Differentiation

Logarithmic differentiation is a method used to differentiate functions that are products or quotients of variables raised to powers, especially when they involve logarithmic functions. By taking the natural logarithm of both sides of an equation, it simplifies the differentiation process, particularly when dealing with exponential growth or decay. This technique is particularly useful for functions that are difficult to differentiate using standard rules.
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Related Practice
Textbook Question

9–61. Evaluate and simplify y'.


y = tan^−1 √t²−1

Textbook Question

Use the given graphs of f and g to find each derivative. <IMAGE>

b. d/dx (f(x)g(x)) |x=1

Textbook Question

9–61. Evaluate and simplify y'.


y = 2^x²−x

Textbook Question

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.

Textbook Question

Use the given graphs of f and g to find each derivative. <IMAGE>

c. d/dx ((f(x) / g(x)) |x=3

Textbook Question

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>


b. d/dx ((f(x) / g(x)) |x=