Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [e^{x^{2}}]$ .

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Step 2.1

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 $g (x) = x^{2}$ .

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

$3 x^{2} + 2 e^{x^{2}} x$

Step 4.2

Reorder factors in $3 x^{2} + 2 e^{x^{2}} x$ .

$3 x^{2} + 2 x e^{x^{2}}$