Textbook Question
Find d/dx(ln√x²+1).
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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 3.4.41
Derivatives Find and simplify the derivative of the following functions.
g(t) = 3t² + 6/t⁷
Verified step by step guidance1
Step 1: Identify the function components. The function is g(t) = 3t^2 + 6/t^7. It consists of two terms: 3t^2 and 6/t^7.
Step 2: Rewrite the function for easier differentiation. The term 6/t^7 can be rewritten using negative exponents as 6t^(-7). So, g(t) = 3t^2 + 6t^(-7).
Step 3: Differentiate each term separately. Use the power rule for differentiation, which states that the derivative of t^n is n*t^(n-1).
Step 4: Apply the power rule to the first term. The derivative of 3t^2 is 2*3*t^(2-1) = 6t.
Step 5: Apply the power rule to the second term. The derivative of 6t^(-7) is -7*6*t^(-7-1) = -42t^(-8).
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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 of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative is often denoted as f'(x) or df/dx, and it can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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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, making it easier to compute derivatives quickly.
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Quotient Rule
The Quotient Rule is a method for differentiating functions that are expressed as the ratio of two other functions. If a function is defined as f(x) = u(x)/v(x), where both u and v are differentiable, the derivative is given by (v*u' - u*v')/v². This rule is essential when dealing with rational functions, allowing for the correct computation of their derivatives.
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Related Practice
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A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?
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Find the slope of the curve y=sin-1 x at (1/2, π/6) without calculating the derivative of sin-1 x.
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