Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 2 - The Derivative - 2.6 The Chain Rule - Exercises Set 2.6: 48

Answer

The equation for the tangent line for $x=\frac{\pi}{4}$ is $$y=-24x+6\pi+3$$

Work Step by Step

The equation for the tangent line at $x_0$ is: $$y-f(x_0)=f'(x_0)(x-x_0)$$ Here is $x_0=\frac{\pi}{4}$ and $f(x)=3\cot^4x$. $$f'(x)=(3\cot^4x)'=3\cdot4\cot^3x(\cot x)'=12\cot^3x\cdot(-\csc^2x)=-12\cot^3x\csc^2x$$ For $x=\frac{\pi}{4}$ we have: $$f(0)=3\cot^3\frac{\pi}{4}=3\cdot1^3=3$$ $$f'(0)=-12\cot^3\frac{\pi}{4}\csc^2\frac{\pi}{4}=-12\cdot1^3\cdot(\sqrt2)^2=-24$$ So the equation for the tangent line for $x=\frac{\pi}{4}$ is: $$y-3=-24(x-\frac{\pi}{4})\Rightarrow y=-24x+6\pi+3$$
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