Textbook Question
9–61. Evaluate and simplify y'.
y = tan^−1 √t²−1
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.12
Evaluate and simplify y'.
y = 2x√2
Verified step by step guidance1
Step 1: Recognize that the function y = 2x^{\(\sqrt{2}\)} is in the form of a power function, where the exponent is an irrational number, \(\sqrt{2}\).
Step 2: To differentiate y with respect to x, use the power rule for differentiation, which states that if y = x^n, then y' = n \(\cdot\) x^{n-1}.
Step 3: Apply the power rule to the given function. Here, n = \(\sqrt{2}\), so the derivative y' = \(\sqrt{2}\) \(\cdot\) 2x^{\(\sqrt{2}\) - 1}.
Step 4: Simplify the expression for y' by multiplying the constant \(\sqrt{2}\) with 2, resulting in y' = 2\(\sqrt{2}\) \(\cdot\) x^{\(\sqrt{2}\) - 1}.
Step 5: The expression y' = 2\(\sqrt{2}\) \(\cdot\) x^{\(\sqrt{2}\) - 1} is the simplified form of the derivative of the given function.
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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. In this case, we need to differentiate the function y = 2x^(√2) to find y'.
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Power Rule
The Power Rule is a basic rule for differentiation that states if y = x^n, then the derivative y' = n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial and power functions, making it essential for evaluating y' in the given function.
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Constant Multiple Rule
The Constant Multiple Rule states that if a function is multiplied by a constant, the derivative of the function is the constant multiplied by the derivative of the function itself. In this case, since y = 2x^(√2), we will apply this rule to factor out the constant 2 when differentiating.
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Related Practice
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