Find the Derivative – d/dx (5x^6+2x^3)^4

${(5 x^{6} + 2 x^{3})}^{4}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 5 x^{6} + 2 x^{3}$ .

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To apply the Chain Rule, set $u$ as $5 x^{6} + 2 x^{3}$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [5 x^{6} + 2 x^{3}]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$4 u^{3} \frac{d}{d x} [5 x^{6} + 2 x^{3}]$

Replace all occurrences of $u$ with $5 x^{6} + 2 x^{3}$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} \frac{d}{d x} [5 x^{6} + 2 x^{3}]$

Differentiate.

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By the Sum Rule, the derivative of $5 x^{6} + 2 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5 x^{6}] + \frac{d}{d x} [2 x^{3}]$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} (\frac{d}{d x} [5 x^{6}] + \frac{d}{d x} [2 x^{3}])$

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{6}$ with respect to $x$ is $5 \frac{d}{d x} [x^{6}]$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} (5 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [2 x^{3}])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} (5 (6 x^{5}) + \frac{d}{d x} [2 x^{3}])$

Multiply $6$ by $5$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} (30 x^{5} + \frac{d}{d x} [2 x^{3}])$

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} (30 x^{5} + 2 \frac{d}{d x} [x^{3}])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} (30 x^{5} + 2 (3 x^{2}))$

Multiply $3$ by $2$ .

$4 {(5 x^{6} + 2 x^{3})}^{3} (30 x^{5} + 6 x^{2})$

Find the Derivative – d/dx (5x^6+2x^3)^4

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