Textbook Question
Evaluate and simplify y'.
y = 2x√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.R.29
9–61. Evaluate and simplify y'.
y = tan^−1 √t²−1
Verified step by step guidance1
First, identify the function y in terms of t. Here, y = tan^(-1)(√(t²−1)).
To find y', the derivative of y with respect to t, use the chain rule. The chain rule states that if a function y = f(g(t)), then y' = f'(g(t)) * g'(t).
Recognize that y = tan^(-1)(u) where u = √(t²−1). The derivative of tan^(-1)(u) with respect to u is 1/(1+u²).
Next, find the derivative of u = √(t²−1) with respect to t. Use the chain rule again: if u = (t²−1)^(1/2), then u' = (1/2)(t²−1)^(-1/2) * 2t = t/√(t²−1).
Combine the derivatives using the chain rule: y' = (1/(1+u²)) * (t/√(t²−1)). Substitute u = √(t²−1) back into the expression to simplify y'.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Inverse Functions
The derivative of an inverse function, such as the arctangent function, can be found using the formula: if y = tan^(-1)(u), then dy/dx = 1/(1 + u^2) * du/dx. This concept is essential for differentiating functions that involve inverse trigonometric functions.
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Derivatives of Inverse Sine & Inverse Cosine
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. If a function y = f(g(t)), the derivative is given by dy/dt = f'(g(t)) * g'(t). This rule is crucial for evaluating the derivative of y = tan^(-1)(√(t² - 1)) since it involves a composition of functions.
Recommended video:
05:02Intro to the Chain Rule
Simplifying Derivatives
After finding the derivative, simplifying the expression is often necessary to make it more manageable or interpretable. This may involve factoring, combining like terms, or rationalizing the denominator. Understanding how to simplify derivatives is key to presenting the final answer clearly.
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6:36Simplifying Trig Expressions
Related Practice
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
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
b. d/dx ((f(x) / g(x)) |x=
Textbook Question
The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>
Compute the average rate of population growth from 1950 to 1960.
Textbook Question
Use differentiation to verify each equation.
d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).