Calculus Examples

$y = (x^{5} + 3) \cdot e^{3 x^{2}}$

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 $f (x) = x^{5} + 3$ and $g (x) = e^{3 x^{2}}$ .

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

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 $g (x) = 3 x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x^{2}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $3 x^{2}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $3$ .

$(x^{5} + 3) e^{3 x^{2}} (6 x) + e^{3 x^{2}} \frac{d}{d x} [x^{5} + 3]$

Step 3.4

By the Sum Rule, the derivative of $x^{5} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [3]$ .

$(x^{5} + 3) e^{3 x^{2}} (6 x) + e^{3 x^{2}} (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [3])$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$(x^{5} + 3) e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4} + \frac{d}{d x} [3])$

Step 3.6

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$(x^{5} + 3) e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4} + 0)$

Step 3.7

Add $5 x^{4}$ and $0$ .

$(x^{5} + 3) e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(x^{5} e^{3 x^{2}} + 3 e^{3 x^{2}}) (6 x) + e^{3 x^{2}} (5 x^{4})$

Step 4.2

Apply the distributive property.

$x^{5} e^{3 x^{2}} (6 x) + 3 e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4})$

Step 4.3

Combine terms.

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Step 4.3.1

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

$x^{1} x^{5} e^{3 x^{2}} \cdot 6 + 3 e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4})$

Step 4.3.2

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

$x^{1 + 5} e^{3 x^{2}} \cdot 6 + 3 e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4})$

Step 4.3.3

Add $1$ and $5$ .

$x^{6} e^{3 x^{2}} \cdot 6 + 3 e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4})$

Step 4.3.4

Move $6$ to the left of $x^{6} e^{3 x^{2}}$ .

$6 \cdot (x^{6} e^{3 x^{2}}) + 3 e^{3 x^{2}} (6 x) + e^{3 x^{2}} (5 x^{4})$

Step 4.3.5

Multiply $6$ by $3$ .

$6 x^{6} e^{3 x^{2}} + 18 e^{3 x^{2}} x + e^{3 x^{2}} (5 x^{4})$

Step 4.4

Reorder terms.

$6 e^{3 x^{2}} x^{6} + 18 e^{3 x^{2}} x + 5 e^{3 x^{2}} x^{4}$