Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
√x⁴+y² = 5x+2y³
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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 3.5.59
Find y'' for the following functions.
y = ex sin x
Verified step by step guidance1
First, identify the function y = e^x * sin(x). This is a product of two functions, so we will use the product rule for differentiation.
Apply the product rule: If y = u * v, then y' = u' * v + u * v'. Here, let u = e^x and v = sin(x).
Differentiate u = e^x to get u' = e^x. Differentiate v = sin(x) to get v' = cos(x).
Substitute these derivatives into the product rule formula: y' = e^x * sin(x) + e^x * cos(x).
Differentiate y' again to find y''. Use the product rule on each term separately: For the first term, differentiate e^x * sin(x) again, and for the second term, differentiate e^x * cos(x). Combine these results to find y''.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Higher-Order Derivatives
Higher-order derivatives refer to the derivatives of a function beyond the first derivative. The second derivative, denoted as y'', measures the rate of change of the first derivative, providing insights into the function's concavity and acceleration. Understanding how to compute higher-order derivatives is essential for analyzing the behavior of functions in calculus.
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02:42
Higher Order Derivatives
Product Rule
The product rule is a fundamental differentiation technique used when finding the derivative of the product of two functions. It states that if u(x) and v(x) are functions, then the derivative of their product is given by u'v + uv'. This rule is particularly important when differentiating functions like y = e^x sin x, which is a product of two distinct functions.
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05:18The Product Rule
Chain Rule
The chain rule is a method for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. This rule is crucial when dealing with functions that involve exponentials and trigonometric functions, as seen in y = e^x sin x.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.
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Textbook Question
Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1
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Textbook Question
15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)
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Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t
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Textbook Question
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(3x + 1)⁴
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