Calculus Examples

$f (x) = {(5 x + 7)}^{2} (e^{x})$

Step 1

Rewrite ${(5 x + 7)}^{2}$ as $(5 x + 7) (5 x + 7)$ .

$\frac{d}{d x} [(5 x + 7) (5 x + 7) e^{x}]$

Step 2

Expand $(5 x + 7) (5 x + 7)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [(5 x (5 x + 7) + 7 (5 x + 7)) e^{x}]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [(5 x (5 x) + 5 x \cdot 7 + 7 (5 x + 7)) e^{x}]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [(5 x (5 x) + 5 x \cdot 7 + 7 (5 x) + 7 \cdot 7) e^{x}]$

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

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [(5 \cdot 5 x \cdot x + 5 x \cdot 7 + 7 (5 x) + 7 \cdot 7) e^{x}]$

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

$\frac{d}{d x} [(5 \cdot 5 (x \cdot x) + 5 x \cdot 7 + 7 (5 x) + 7 \cdot 7) e^{x}]$

Step 3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [(5 \cdot 5 x^{2} + 5 x \cdot 7 + 7 (5 x) + 7 \cdot 7) e^{x}]$

Step 3.1.3

Multiply $5$ by $5$ .

$\frac{d}{d x} [(25 x^{2} + 5 x \cdot 7 + 7 (5 x) + 7 \cdot 7) e^{x}]$

Step 3.1.4

Multiply $7$ by $5$ .

$\frac{d}{d x} [(25 x^{2} + 35 x + 7 (5 x) + 7 \cdot 7) e^{x}]$

Step 3.1.5

Multiply $5$ by $7$ .

$\frac{d}{d x} [(25 x^{2} + 35 x + 35 x + 7 \cdot 7) e^{x}]$

Step 3.1.6

Multiply $7$ by $7$ .

$\frac{d}{d x} [(25 x^{2} + 35 x + 35 x + 49) e^{x}]$

Step 3.2

Add $35 x$ and $35 x$ .

$\frac{d}{d x} [(25 x^{2} + 70 x + 49) e^{x}]$

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 $f (x) = 25 x^{2} + 70 x + 49$ and $g (x) = e^{x}$ .

$(25 x^{2} + 70 x + 49) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [25 x^{2} + 70 x + 49]$

Step 5

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

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} \frac{d}{d x} [25 x^{2} + 70 x + 49]$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $25 x^{2} + 70 x + 49$ with respect to $x$ is $\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [70 x] + \frac{d}{d x} [49]$ .

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [70 x] + \frac{d}{d x} [49])$

Step 6.2

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

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (25 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [70 x] + \frac{d}{d x} [49])$

Step 6.3

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

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (25 (2 x) + \frac{d}{d x} [70 x] + \frac{d}{d x} [49])$

Step 6.4

Multiply $2$ by $25$ .

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (50 x + \frac{d}{d x} [70 x] + \frac{d}{d x} [49])$

Step 6.5

Since $70$ is constant with respect to $x$ , the derivative of $70 x$ with respect to $x$ is $70 \frac{d}{d x} [x]$ .

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (50 x + 70 \frac{d}{d x} [x] + \frac{d}{d x} [49])$

Step 6.6

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

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (50 x + 70 \cdot 1 + \frac{d}{d x} [49])$

Step 6.7

Multiply $70$ by $1$ .

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (50 x + 70 + \frac{d}{d x} [49])$

Step 6.8

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (50 x + 70 + 0)$

Step 6.9

Add $50 x + 70$ and $0$ .

$(25 x^{2} + 70 x + 49) e^{x} + e^{x} (50 x + 70)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$25 x^{2} e^{x} + 70 x e^{x} + 49 e^{x} + e^{x} (50 x + 70)$

Step 7.2

Apply the distributive property.

$25 x^{2} e^{x} + 70 x e^{x} + 49 e^{x} + e^{x} (50 x) + e^{x} \cdot 70$

Step 7.3

Combine terms.

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Step 7.3.1

Move $70$ to the left of $e^{x}$ .

$25 x^{2} e^{x} + 70 x e^{x} + 49 e^{x} + e^{x} (50 x) + 70 \cdot e^{x}$

Step 7.3.2

Add $70 x e^{x}$ and $e^{x} (50 x)$ .

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Step 7.3.2.1

Move $e^{x}$ .

$25 x^{2} e^{x} + 70 x e^{x} + 50 x e^{x} + 49 e^{x} + 70 e^{x}$

Step 7.3.2.2

Add $70 x e^{x}$ and $50 x e^{x}$ .

$25 x^{2} e^{x} + 120 x e^{x} + 49 e^{x} + 70 e^{x}$

Step 7.3.3

Add $49 e^{x}$ and $70 e^{x}$ .

$25 x^{2} e^{x} + 120 x e^{x} + 119 e^{x}$

Step 7.4

Reorder terms.

$25 e^{x} x^{2} + 120 e^{x} x + 119 e^{x}$

Step 7.5

Reorder factors in $25 e^{x} x^{2} + 120 e^{x} x + 119 e^{x}$ .

$25 x^{2} e^{x} + 120 x e^{x} + 119 e^{x}$