Calculus Examples

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

Step 1

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 $g (x) = e^{x^{3}}$ .

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

Step 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 $g (x) = x^{3}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Move $x^{2}$ .

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

Step 4.2

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

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

Step 4.3

Add $2$ and $2$ .

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

Step 5

Move $3$ to the left of $e^{x^{3}}$ .

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Reorder terms.

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

Step 7.2

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

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