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Find the following derivatives. Do not simplify your answers. d sin (3x) dx e²x + x² a)

Find the following derivatives. Do not simplify your answers. d sin (3x) dx e²x + x² a)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
2. Find the following derivatives. Do not simplify your answers. d [sin (3x) dx e²x + x² a) b) [√tsec²(t²) - e³x] dt
Transcribed Image Text:2. Find the following derivatives. Do not simplify your answers. d [sin (3x) dx e²x + x² a) b) [√tsec²(t²) - e³x] dt
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\frac{d}{dx}\left(\frac{\sin \left(3x\right)}{e^{2x}+x^2}\right)

use the quotient rule for the derivative

\left(\frac{f}{g}\right)'=\frac{f\:'\cdot g-g'\cdot f}{g^2}

=\frac{\frac{d}{dx}\left(\sin \left(3x\right)\right)\left(e^{2x}+x^2\right)-\frac{d}{dx}\left(e^{2x}+x^2\right)\sin \left(3x\right)}{\left(e^{2x}+x^2\right)^2}

=\frac{3\cos \left(3x\right)\left(e^{2x}+x^2\right)-\frac{d}{dx}\left(e^{2x}+x^2\right)\sin \left(3x\right)}{\left(e^{2x}+x^2\right)^2}

=\frac{3\cos \left(3x\right)\left(e^{2x}+x^2\right)-\left(\frac{d}{dx}\left(e^{2x}\right)+\frac{d}{dx}\left(x^2\right)\right)\sin \left(3x\right)}{\left(e^{2x}+x^2\right)^2}

=\frac{3\cos \left(3x\right)\left(e^{2x}+x^2\right)-\left(e^{2x}\cdot \:\:2+2x\right)\sin \left(3x\right)}{\left(e^{2x}+x^2\right)^2}

{\color{Red} =\frac{3\cos \left(3x\right)\left(e^{2x}+x^2\right)-\left(2e^{2x}+2x\right)\sin \left(3x\right)}{\left(e^{2x}+x^2\right)^2}}

NOTE: we do not have to simplify the answer

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