Calculus Examples

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

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

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

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 $x^{3}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

$x^{2} x^{1} (e^{x^{3}} \cdot (3)) + e^{x^{3}} \frac{d}{d x} [x]$

Step 5

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

$x^{2 + 1} (e^{x^{3}} \cdot (3)) + e^{x^{3}} \frac{d}{d x} [x]$

Step 6

Simplify the expression.

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Step 6.1

Add $2$ and $1$ .

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

Step 6.2

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

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

Step 7

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

$x^{3} (3 e^{x^{3}}) + e^{x^{3}} \cdot 1$

Step 8

Multiply $e^{x^{3}}$ by $1$ .

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

Step 9

Simplify.

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Step 9.1

Reorder terms.

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

Step 9.2

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

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