David Lewis

2022-01-15

Derivatives involving ln x Find the following derivatives.

$\frac{d}{dx}\left(\mathrm{ln}2{x}^{8}\right)$

abonirali59

Beginner2022-01-16Added 35 answers

Step 1

Given:

$\frac{d}{dx}\left(\mathrm{ln}2{x}^{8}\right)$

Step 2

$\frac{d}{dx}\left(\mathrm{ln}\left(2{x}^{8}\right)\right)$

Apply chain rule:$\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)=\frac{d}{du}f\left(u\right)\frac{d}{dx}g\left(x\right)$ with $u=2{x}^{8}$

$=\frac{d}{du}\mathrm{ln}\left(u\right)\frac{d}{dx}\left(2{x}^{8}\right)$

Apply power rule:$\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$

$=\frac{1}{u}\times 2\cdot 8{x}^{8-1}$

$=\frac{16{x}^{7}}{u}$

Put back:$u=2{x}^{8}$

$=\frac{16{x}^{7}}{2{x}^{8}}$

$=\frac{8}{x}$

Given:

Step 2

Apply chain rule:

Apply power rule:

Put back:

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