Calculus Examples

$y = e^{x^{3} + 8 x^{2}}$

Step 1

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 $g (x) = x^{3} + 8 x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{3} + 8 x^{2}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{3} + 8 x^{2}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [x^{3} + 8 x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $x^{3} + 8 x^{2}$ .

$e^{x^{3} + 8 x^{2}} \frac{d}{d x} [x^{3} + 8 x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + 8 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 x^{2}]$ .

$e^{x^{3} + 8 x^{2}} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 x^{2}])$

Step 2.2

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

$e^{x^{3} + 8 x^{2}} (3 x^{2} + \frac{d}{d x} [8 x^{2}])$

Step 2.3

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$e^{x^{3} + 8 x^{2}} (3 x^{2} + 8 \frac{d}{d x} [x^{2}])$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{x^{3} + 8 x^{2}} (3 x^{2} + 8 (2 x))$

Step 2.5

Multiply $2$ by $8$ .

$e^{x^{3} + 8 x^{2}} (3 x^{2} + 16 x)$