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fast = y = e²xsin(5x)

fast = y = e²xsin(5x)

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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**Problem Statement:** Find the first and second derivatives of the function. \[ y = e^{2x} \sin(5x) \] **First Derivative:** \[ y' = \] **Second Derivative:** \[ y'' = \] **Explanation:** The problem asks to determine the derivatives of the given function. The function consists of an exponential part, \( e^{2x} \), and a trigonometric part, \( \sin(5x) \). To solve this, one can use the product rule for differentiation, along with the chain rule where necessary, to find the derivatives. - **The product rule** for derivatives states that if you have two functions multiplied together, \( u(x) \) and \( v(x) \), their derivative \( (uv)' \) is given by: \[ (uv)' = u'v + uv' \] - **The chain rule** is used for differentiating compositions of functions, allowing the derivative of an outer function to multiply by the derivative of the inner function.
Transcribed Image Text:**Problem Statement:** Find the first and second derivatives of the function. \[ y = e^{2x} \sin(5x) \] **First Derivative:** \[ y' = \] **Second Derivative:** \[ y'' = \] **Explanation:** The problem asks to determine the derivatives of the given function. The function consists of an exponential part, \( e^{2x} \), and a trigonometric part, \( \sin(5x) \). To solve this, one can use the product rule for differentiation, along with the chain rule where necessary, to find the derivatives. - **The product rule** for derivatives states that if you have two functions multiplied together, \( u(x) \) and \( v(x) \), their derivative \( (uv)' \) is given by: \[ (uv)' = u'v + uv' \] - **The chain rule** is used for differentiating compositions of functions, allowing the derivative of an outer function to multiply by the derivative of the inner function.
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Step 1: Given and Required

We have given a function

                     y space equals space space 2 e to the power of 2 x end exponent space sin left parenthesis 5 x right parenthesis

We need to find the first and second derivatives of a function.

i) y'

ii) y''

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