Best solutions to - "Find the derivative of x 3 ln ⁡..." - Plainmath

Jordin Olsen

Answered question

2022-05-01

Find the derivative of

\frac{x^{3}}{\ln (x)}

.

Answer & Explanation

Brennen Davies

Beginner2022-05-02Added 25 answers

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{g (x)^{2}}

where

f (x) = x^{3}

and

g (x) = \ln (x)

\frac{\ln (x) \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

is

n x^{n - 1}

where n=3.

\frac{\ln (x) (3 x^{2}) - x^{3} \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}

The derivative of

\ln (x)

with respect to x is

\frac{1}{x}

\frac{\ln (x) (3 x^{2}) - x^{3} \frac{1}{x}}{\ln^{2} (x)}

Simplify.

- \frac{x^{2} (1 - \ln (x^{3}))}{\ln^{2} (x)}

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