What is the derivative of ln ⁡ ( 1 + 4 x ) 2 + 6 x 3 ?

Question

What is the derivative of             ln      ⁡      (      1      +      4      x      )              2      +      6              x                  3                    ?

Darion Sexton · 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 $f (x) = \ln (1 + 4 x)$ and $g (x) = 2 + 6 x^{3}$ .

$\frac{(2 + 6 x^{3}) \frac{d}{d x} [\ln (1 + 4 x)] - \ln (1 + 4 x) \frac{d}{d x} [2 + 6 x^{3}]}{{(2 + 6 x^{3})}^{2}}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 1 + 4 x$ .

$\frac{(2 + 6 x^{3}) (\frac{1}{1 + 4 x} \frac{d}{d x} [1 + 4 x]) - \ln (1 + 4 x) \frac{d}{d x} [2 + 6 x^{3}]}{{(2 + 6 x^{3})}^{2}}$

Differentiate.

$\frac{(2 + 6 x^{3}) \frac{4}{1 + 4 x} - 18 \ln (1 + 4 x) x^{2}}{{(2 + 6 x^{3})}^{2}}$

Simplify.

$\frac{24 x^{3} - x^{2} \ln ({(1 + 4 x)}^{18}) - x^{3} \ln ({(1 + 4 x)}^{72}) + 8}{{(2 + 6 x^{3})}^{2} (1 + 4 x)}$

What is the derivative of <mrow> ln ⁡<!-- ⁡ 1 +

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