Calculus Examples

$2 x e^{5 x^{3}}$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 x e^{5 x^{3}}$ with respect to $x$ is $2 \frac{d}{d x} [x e^{5 x^{3}}]$ .

$2 \frac{d}{d x} [x e^{5 x^{3}}]$

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 $f (x) = x$ and $g (x) = e^{5 x^{3}}$ .

$2 (x \frac{d}{d x} [e^{5 x^{3}}] + e^{5 x^{3}} \frac{d}{d x} [x])$

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 $g (x) = 5 x^{3}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $5 x^{3}$ .

$2 (x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x^{3}]) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 3.2

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

$2 (x (e^{u} \frac{d}{d x} [5 x^{3}]) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 3.3

Replace all occurrences of $u$ with $5 x^{3}$ .

$2 (x (e^{5 x^{3}} \frac{d}{d x} [5 x^{3}]) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 4

Differentiate.

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$2 (x (e^{5 x^{3}} (5 \frac{d}{d x} [x^{3}])) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 4.2

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

$2 (x (e^{5 x^{3}} (5 (3 x^{2}))) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 4.3

Multiply $3$ by $5$ .

$2 (x (e^{5 x^{3}} (15 x^{2})) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 5

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

$2 (x^{2} x^{1} (e^{5 x^{3}} \cdot (15)) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 6

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

$2 (x^{2 + 1} (e^{5 x^{3}} \cdot (15)) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 7

Simplify the expression.

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Step 7.1

Add $2$ and $1$ .

$2 (x^{3} (e^{5 x^{3}} \cdot (15)) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 7.2

Move $15$ to the left of $e^{5 x^{3}}$ .

$2 (x^{3} (15 \cdot e^{5 x^{3}}) + e^{5 x^{3}} \frac{d}{d x} [x])$

Step 8

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

$2 (x^{3} (15 e^{5 x^{3}}) + e^{5 x^{3}} \cdot 1)$

Step 9

Multiply $e^{5 x^{3}}$ by $1$ .

$2 (x^{3} (15 e^{5 x^{3}}) + e^{5 x^{3}})$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$2 (x^{3} (15 e^{5 x^{3}})) + 2 e^{5 x^{3}}$

Step 10.2

Multiply $15$ by $2$ .

$30 x^{3} e^{5 x^{3}} + 2 e^{5 x^{3}}$

Step 10.3

Reorder terms.

$30 e^{5 x^{3}} x^{3} + 2 e^{5 x^{3}}$

Step 10.4

Reorder factors in $30 e^{5 x^{3}} x^{3} + 2 e^{5 x^{3}}$ .

$30 x^{3} e^{5 x^{3}} + 2 e^{5 x^{3}}$