What is the derivative of ln ⁡ ( x ) x 2 ?

Differentiate using the Quotient Rule which states that d d x [ f ( x ) g ( x ) ] is g ( x ) d d x [ f ( x ) ] − f ( x ) d d x [ g ( x ) ] g ( x ) 2 f ( x ) = ln ⁡ ( x ) and g ( x ) = x 2 x 2 d d x [ ln ⁡ ( x ) ] − ln ⁡ ( x ) d d x [ x 2 ] ( x 2 ) 2 The derivative of ln ⁡ ( x ) with respect to x is 1 x x 2 1 x − ln ⁡ ( x ) d d x [ x 2 ] ( x 2 ) 2 Differentiate using the Power Rule which states that d d x [ x n ] is n x n − 1 where n=2. x 2 1 x − ln ⁡ ( x ) ( 2 x ) ( x 2 ) 2 Simplify. 1 − ln ⁡ ( x 2 ) x 3

Step by step solution to "What is the derivative of ln ⁡ (..."

Esther Hoffman

Answered question

2022-05-01

What is the derivative of

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

Answer & Explanation

Colonninisxi

Beginner2022-05-02Added 16 answers

Differentiate using the Quotient Rule which states that

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

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

f (x) = \ln (x)

and

g (x) = x^{2}

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

The derivative of

\ln (x)

with respect to x is

\frac{1}{x}

\frac{x^{2} \frac{1}{x} - \ln (x) \frac{d}{d x} [x^{2}]}{(x^{2})^{2}}

Differentiate using the Power Rule which states that

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

n x^{n - 1}

where n=2.

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

Simplify.

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

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