Check the solution to "Derivatives involving ln x Find the following derivatives...."

David Lewis

Answered question

2022-01-15

Derivatives involving ln x Find the following derivatives.

\frac{d}{d x} (\ln 2 x^{8})

abonirali59

Beginner2022-01-16Added 35 answers

Step 1
Given:

\frac{d}{d x} (\ln 2 x^{8})

Step 2

\frac{d}{d x} (\ln (2 x^{8}))

Apply chain rule:

\frac{d}{d x} (f (g (x))) = \frac{d}{d u} f (u) \frac{d}{d x} g (x)

with

u = 2 x^{8}

= \frac{d}{d u} \ln (u) \frac{d}{d x} (2 x^{8})

Apply power rule:

\frac{d}{d x} (x^{n}) = 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}

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