Calculus Examples

$f (x) = x^{2} (2 e^{x^{2}})$

Step 1

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = e^{x^{2}}$ .

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

Step 3

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 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.1

Move $x$ .

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

Step 5.2

Multiply $x$ by $x^{2}$ .

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Step 5.2.1

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{2} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{2}])$

Step 5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 (x^{1 + 2} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{2}])$

Step 5.3

Add $1$ and $2$ .

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

Step 6

Move $2$ to the left of $e^{x^{2}}$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Combine terms.

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Step 8.2.1

Multiply $2$ by $2$ .

$4 (x^{3} (e^{x^{2}})) + 2 (e^{x^{2}} (2 x))$

Step 8.2.2

Multiply $2$ by $2$ .

$4 x^{3} e^{x^{2}} + 4 e^{x^{2}} x$

Step 8.3

Reorder terms.

$4 e^{x^{2}} x^{3} + 4 e^{x^{2}} x$

Step 8.4

Reorder factors in $4 e^{x^{2}} x^{3} + 4 e^{x^{2}} x$ .

$4 x^{3} e^{x^{2}} + 4 x e^{x^{2}}$