Find the derivative of e 1 + 4 x 4 + 6 x 3 .

Question

Find the derivative of             e              1        +        4        x                    4      +      6              x                  3                    .

kazneni3tr2b · 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 $f (x) = e^{1 + 4 x}$ and $g (x) = 4 + 6 x^{3}$ .

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

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

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

Differentiate.

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

Simplify.

$\frac{e^{1 + 4 x} (12 x^{3} - 9 x^{2} + 8)}{2 {(3 x^{3} + 2)}^{2}}$

Find the derivative of e 1 + 4 x </mrow>

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