Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.1 - Derivatives of Polynomials and Exponential Functions - 3.1 Exercises - Page 180: 48


$f(x)=e^x-x^3$ $f'(x)=e^x-3x^2$ $f''(x)=e^x-6x$

Work Step by Step

Let's first remember the power rule: $f(x)=x^n$ $f'(x)=nx^{n-1}$ And we also know that $\frac{d}{dx}[e^x]=e^x$ So with this information, we can find the first and second derivatives. We know that $f(x)=e^x-x^3$ So by using the power rule and the differentiation of an exponential function, we can find that: $f'(x)=e^x-3x^2$ And by taking the derivative of that, we can find the second derivative: $f''(x)=e^x-6x$ So now you now that $f(x)=e^x-x^3$ $f'(x)=e^x-3x^2$ $f''(x)=e^x-6x$ You can graph each equation together to make sure these answers are reasonable.
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