Take the derivative of 2 + 2 x + x 2 ln ⁡ ( x ) .

Question

Take the derivative of             2      +      2      x      +              x                  2                            ln      ⁡      (      x      )      .

wellnesshaus4n4 · Accepted Answer

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)=2+2x+x and

g (x) = \ln (x)

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

Differentiate.

\frac{\ln (x) (0 + \frac{d}{d x} [2 x] + \frac{d}{d x} [x^{2}]) - (2 + 2 x + x^{2}) \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}

Evaluate

\frac{d}{d x} [2 x]

\frac{\ln (x) (0 + 2 + \frac{d}{d x} [x^{2}]) - (2 + 2 x + x^{2}) \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=2.

\frac{\ln (x) (0 + 2 + 2 x) - (2 + 2 x + x^{2}) \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}

Simplify.

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

Simplify.

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

Take the derivative of 2 + 2 x + x <mr

Answered question

Answer & Explanation

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