Calculus Examples

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

Step 1

Rewrite ${(3 + 5 x)}^{2}$ as $(3 + 5 x) (3 + 5 x)$ .

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

Step 2

Expand $(3 + 5 x) (3 + 5 x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $3$ by $3$ .

$\frac{d}{d x} [e^{x} (9 + 3 (5 x) + 5 x \cdot 3 + 5 x (5 x))]$

Step 3.1.2

Multiply $5$ by $3$ .

$\frac{d}{d x} [e^{x} (9 + 15 x + 5 x \cdot 3 + 5 x (5 x))]$

Step 3.1.3

Multiply $3$ by $5$ .

$\frac{d}{d x} [e^{x} (9 + 15 x + 15 x + 5 x (5 x))]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [e^{x} (9 + 15 x + 15 x + 5 \cdot 5 x \cdot x)]$

Step 3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [e^{x} (9 + 15 x + 15 x + 5 \cdot 5 (x \cdot x))]$

Step 3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [e^{x} (9 + 15 x + 15 x + 5 \cdot 5 x^{2})]$

Step 3.1.6

Multiply $5$ by $5$ .

$\frac{d}{d x} [e^{x} (9 + 15 x + 15 x + 25 x^{2})]$

Step 3.2

Add $15 x$ and $15 x$ .

$\frac{d}{d x} [e^{x} (9 + 30 x + 25 x^{2})]$

Step 4

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 $f (x) = e^{x}$ and $g (x) = 9 + 30 x + 25 x^{2}$ .

$e^{x} \frac{d}{d x} [9 + 30 x + 25 x^{2}] + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $9 + 30 x + 25 x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [30 x] + \frac{d}{d x} [25 x^{2}]$ .

$e^{x} (\frac{d}{d x} [9] + \frac{d}{d x} [30 x] + \frac{d}{d x} [25 x^{2}]) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$e^{x} (0 + \frac{d}{d x} [30 x] + \frac{d}{d x} [25 x^{2}]) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.3

Add $0$ and $\frac{d}{d x} [30 x]$ .

$e^{x} (\frac{d}{d x} [30 x] + \frac{d}{d x} [25 x^{2}]) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.4

Since $30$ is constant with respect to $x$ , the derivative of $30 x$ with respect to $x$ is $30 \frac{d}{d x} [x]$ .

$e^{x} (30 \frac{d}{d x} [x] + \frac{d}{d x} [25 x^{2}]) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.5

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

$e^{x} (30 \cdot 1 + \frac{d}{d x} [25 x^{2}]) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.6

Multiply $30$ by $1$ .

$e^{x} (30 + \frac{d}{d x} [25 x^{2}]) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.7

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x^{2}]$ .

$e^{x} (30 + 25 \frac{d}{d x} [x^{2}]) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.8

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

$e^{x} (30 + 25 (2 x)) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 5.9

Multiply $2$ by $25$ .

$e^{x} (30 + 50 x) + (9 + 30 x + 25 x^{2}) \frac{d}{d x} [e^{x}]$

Step 6

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

$e^{x} (30 + 50 x) + (9 + 30 x + 25 x^{2}) e^{x}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$e^{x} \cdot 30 + e^{x} (50 x) + (9 + 30 x + 25 x^{2}) e^{x}$

Step 7.2

Apply the distributive property.

$e^{x} \cdot 30 + e^{x} (50 x) + 9 e^{x} + 30 x e^{x} + 25 x^{2} e^{x}$

Step 7.3

Combine terms.

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Step 7.3.1

Move $30$ to the left of $e^{x}$ .

$30 \cdot e^{x} + e^{x} (50 x) + 9 e^{x} + 30 x e^{x} + 25 x^{2} e^{x}$

Step 7.3.2

Add $30 e^{x}$ and $9 e^{x}$ .

$e^{x} (50 x) + 39 e^{x} + 30 x e^{x} + 25 x^{2} e^{x}$

Step 7.3.3

Add $e^{x} (50 x)$ and $30 x e^{x}$ .

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Step 7.3.3.1

Move $e^{x}$ .

$50 x e^{x} + 30 x e^{x} + 39 e^{x} + 25 x^{2} e^{x}$

Step 7.3.3.2

Add $50 x e^{x}$ and $30 x e^{x}$ .

$80 x e^{x} + 39 e^{x} + 25 x^{2} e^{x}$

Step 7.4

Reorder terms.

$80 e^{x} x + 25 e^{x} x^{2} + 39 e^{x}$

Step 7.5

Reorder factors in $80 e^{x} x + 25 e^{x} x^{2} + 39 e^{x}$ .

$80 x e^{x} + 25 x^{2} e^{x} + 39 e^{x}$