Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 2 - Differentiation - 2.2 Exercises: 27

Answer

Rewrite:$(\frac{6}{125})(x^{-3})$; Differentiate: $(\frac{6}{125})(-3)(x^{-3-1})$; Simplify: $\frac{-18}{125x^4}$

Work Step by Step

To simplify, you have to make sure to notice that not only $x$ is raised to to the power of $3$, it is $5x$ that is raised; hence the denominator becomes $125x^{3}$. Then, you have to take $x^{3}$ to the numerator by flipping the sign of the power; hence we get :$(\frac{6}{125})(x^{-3})$. To differentiate, just use the power rule to find the derivative of $x^{-3}$ which is $(-3)(x^{-3-1})$. The overall derivative is the derivative of $x^{-3}$ times the constant $\frac{6}{125}$. Hence we get $(\frac{6}{125})(-3)(x^{-3-1})$. To simplify, you have to take the $x^{-4}$ to the denominator and flipping the exponent's sign (Index rule: $x^{-n}=\frac{1}{x^n}$ ). Hence the overall final derivative is $\frac{-18}{125x^4}$.
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