Find out the answer to "Derivatives Evaluate the following derivatives. ddx((ln⁡2x)−5)"

Gregory Jones

Answered question

2022-01-15

Derivatives Evaluate the following derivatives.

\frac{d}{d x} ({(\ln 2 x)}^{- 5})

aquariump9

Beginner2022-01-16Added 40 answers

Step 1
Given:

\frac{d}{d x} ({(\ln 2 x)}^{- 5})

We have to find derivative of f(x) with respective x.
Step 2

f^{'} (x) = \frac{d {(\ln (2 x))}^{- 5}}{d x}

By the chain rule,

\frac{d f (g (x))}{d x} = \frac{d f (g (x))}{d (g (x))} \frac{d (g (x))}{d x} = f^{'} (g (x)) \cdot g^{'} (x)

f^{'} (x) = - 5 {(\ln (2 x))}^{- 5 - 1} \frac{1}{2 x} 2

= - 5 {(\ln (2 x))}^{- 6} \frac{1}{x}

= \frac{- 5}{x {(\ln (2 x))}^{6}}

Hence the derivative of

{(\ln (2 x))}^{- 5}

with respective x is

\frac{- 5}{x {(\ln (2 x))}^{6}}

Bukvald5z

Beginner2022-01-17Added 33 answers

We are going to use
Power rule:

{(x^{n})}^{'} = n x^{n - 1}

Chain rule: (f*g)'(x)=f'(g(x))*g'(x)
Now

{[{(\ln 2 x)}^{- 5}]}^{'} = - 5 {(\ln 2 x)}^{- 6} \cdot {(\ln 2 x)}^{'} = - 5 {(\ln 2 x)}^{- 6} \cdot \frac{1}{2 x} \cdot 2 =

= - \frac{5}{x} {(\ln 2 x)}^{- 6}

Result:

- \frac{5}{x} {(\ln 2 x)}^{- 6}

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