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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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 3.10.40
13–40. Evaluate the derivative of the following functions.
f(x) = 1/tan^−1(x²+4)
Verified step by step guidance1
Step 1: Recognize that the function f(x) = \(\frac{1}{\tan^{-1}\)(x^2 + 4)} is a composition of functions, where the outer function is g(u) = \(\frac{1}{u}\) and the inner function is u(x) = \(\tan\)^{-1}(x^2 + 4).
Step 2: Apply the chain rule for differentiation, which states that the derivative of a composite function g(u(x)) is g'(u(x)) * u'(x).
Step 3: Differentiate the outer function g(u) = \(\frac{1}{u}\) with respect to u to get g'(u) = -\(\frac{1}{u^2}\).
Step 4: Differentiate the inner function u(x) = \(\tan\)^{-1}(x^2 + 4) with respect to x. Use the derivative formula for \(\tan\)^{-1}(v), which is \(\frac{1}{1+v^2}\), and apply it to v = x^2 + 4. Also, apply the chain rule to differentiate x^2 + 4.
Step 5: Combine the results from Steps 3 and 4 using the chain rule: f'(x) = g'(u(x)) * u'(x). Substitute back u(x) = \(\tan\)^{-1}(x^2 + 4) and simplify the expression.
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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 that provides the slope of the tangent line to the function's graph at any given point. The derivative is often denoted as f'(x) or df/dx and can be calculated using various rules such as the power rule, product rule, and chain rule.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(u), where u = g(x), then the derivative of the composite function f(g(x)) is f'(g(x)) * g'(x). This rule is essential when differentiating functions that involve nested expressions, such as the function in the given question.
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Inverse Trigonometric Functions
Inverse trigonometric functions, such as tan^−1(x), are the functions that reverse the action of the standard trigonometric functions. For example, tan^−1(x) gives the angle whose tangent is x. When differentiating functions involving inverse trigonometric functions, specific derivative formulas apply, such as the derivative of tan^−1(x) being 1/(1+x²), which is crucial for evaluating the derivative of the given function.
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Related Practice
Textbook Question
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
Graph the following curves and determine the location of any vertical tangent lines.
a. x²+y² = 9
Textbook Question
If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.
Textbook Question
Find d²/dx² (sin x + cos x).
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
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
f(x) = 3x⁴(2x²−1)
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