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.48
9–61. Evaluate and simplify y'.
y = 10^sin x+sin¹⁰x
Verified step by step guidance1
Step 1: Identify the function y = 10^{\(\sin\) x} + (\(\sin\) x)^{10}. This function is a sum of two terms: 10^{\(\sin\) x} and (\(\sin\) x)^{10}.
Step 2: Differentiate the first term 10^{\(\sin\) x} using the chain rule. The derivative of a^u, where a is a constant and u is a function of x, is a^u \(\ln\)(a) \(\cdot\) u'. Here, a = 10 and u = \(\sin\) x.
Step 3: Differentiate \(\sin\) x with respect to x to find u'. The derivative of \(\sin\) x is \(\cos\) x.
Step 4: Differentiate the second term (\(\sin\) x)^{10} using the chain rule. The derivative of u^n, where u is a function of x and n is a constant, is n \(\cdot\) u^{n-1} \(\cdot\) u'. Here, u = \(\sin\) x and n = 10.
Step 5: Combine the derivatives from Steps 2 and 4 to find y'. The derivative of y is the sum of the derivatives of its terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the given function y = 10^sin x + sin¹⁰x to find y'.
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Chain Rule
The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is particularly relevant for terms like 10^sin x.
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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 will be applied to the term sin¹⁰x, where we treat sin x as a function raised to a power, allowing us to find its derivative effectively.
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Related Practice
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Evaluate and simplify y'.
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Textbook Question
9–61. Evaluate and simplify y'.
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Textbook Question
Evaluate and simplify y'.
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