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Q. 37

Expert-verified
Found in: Page 210

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating. $f\left(x\right)={\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{100}$

The required answer is $100{\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{99}\left(180{x}^{3}{\left(3{x}^{4}-1\right)}^{2}+3\right)$

See the step by step solution

## Step 1. Given Information

The given function is $f\left(x\right)={\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{100}$

## Step 2. Calculation

Differentiate both the sides with respect to x, we get,

$f\text{'}\left(x\right)=\frac{d}{dx}{\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{100}\phantom{\rule{0ex}{0ex}}=100{\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{100-1}\left(\frac{d}{dx}\left(5{\left(3{x}^{4}-1\right)}^{3}\right)+\frac{d}{dx}\left(3x\right)-\frac{d}{dx}1\right)\phantom{\rule{0ex}{0ex}}=100{\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{99}\left(5\left(3{\left(3{x}^{4}-1\right)}^{3-1}\right)\frac{d}{dx}\left(3{x}^{4}-1\right)+3\right)\phantom{\rule{0ex}{0ex}}=100{\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{99}\left(15{\left(3{x}^{4}-1\right)}^{2}\right)\left(12{x}^{3}\right)+3\right)\phantom{\rule{0ex}{0ex}}=100{\left(5{\left(3{x}^{4}-1\right)}^{3}+3x-1\right)}^{99}\left(180{x}^{3}{\left(3{x}^{4}-1\right)}^{2}+3\right)$