Calculus Examples

$y = {(x + 3)}^{2} e^{4 x}$

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $x$ by $x$ .

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

Step 3.1.2

Move $3$ to the left of $x$ .

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

Step 3.1.3

Multiply $3$ by $3$ .

$\frac{d}{d x} [(x^{2} + 3 x + 3 x + 9) e^{4 x}]$

Step 3.2

Add $3 x$ and $3 x$ .

$\frac{d}{d x} [(x^{2} + 6 x + 9) e^{4 x}]$

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 $f (x) = x^{2} + 6 x + 9$ and $g (x) = e^{4 x}$ .

$(x^{2} + 6 x + 9) \frac{d}{d x} [e^{4 x}] + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 5

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 $g (x) = 4 x$ .

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Step 5.1

To apply the Chain Rule, set $u$ as $4 x$ .

$(x^{2} + 6 x + 9) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [4 x]) + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 5.2

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

$(x^{2} + 6 x + 9) (e^{u} \frac{d}{d x} [4 x]) + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 5.3

Replace all occurrences of $u$ with $4 x$ .

$(x^{2} + 6 x + 9) (e^{4 x} \frac{d}{d x} [4 x]) + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 6

Differentiate.

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Step 6.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$(x^{2} + 6 x + 9) e^{4 x} (4 \frac{d}{d x} [x]) + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 6.2

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

$(x^{2} + 6 x + 9) e^{4 x} (4 \cdot 1) + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $4$ by $1$ .

$(x^{2} + 6 x + 9) e^{4 x} \cdot 4 + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 6.3.2

Move $4$ to the left of $(x^{2} + 6 x + 9) e^{4 x}$ .

$4 \cdot ((x^{2} + 6 x + 9) e^{4 x}) + e^{4 x} \frac{d}{d x} [x^{2} + 6 x + 9]$

Step 6.4

By the Sum Rule, the derivative of $x^{2} + 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [9]$ .

$4 (x^{2} + 6 x + 9) e^{4 x} + e^{4 x} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [9])$

Step 6.5

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

$4 (x^{2} + 6 x + 9) e^{4 x} + e^{4 x} (2 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [9])$

Step 6.6

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$4 (x^{2} + 6 x + 9) e^{4 x} + e^{4 x} (2 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 6.7

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

$4 (x^{2} + 6 x + 9) e^{4 x} + e^{4 x} (2 x + 6 \cdot 1 + \frac{d}{d x} [9])$

Step 6.8

Multiply $6$ by $1$ .

$4 (x^{2} + 6 x + 9) e^{4 x} + e^{4 x} (2 x + 6 + \frac{d}{d x} [9])$

Step 6.9

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

$4 (x^{2} + 6 x + 9) e^{4 x} + e^{4 x} (2 x + 6 + 0)$

Step 6.10

Add $2 x + 6$ and $0$ .

$4 (x^{2} + 6 x + 9) e^{4 x} + e^{4 x} (2 x + 6)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$(4 x^{2} + 4 (6 x) + 4 \cdot 9) e^{4 x} + e^{4 x} (2 x + 6)$

Step 7.2

Apply the distributive property.

$4 x^{2} e^{4 x} + 4 (6 x) e^{4 x} + 4 \cdot 9 e^{4 x} + e^{4 x} (2 x + 6)$

Step 7.3

Apply the distributive property.

$4 x^{2} e^{4 x} + 4 (6 x) e^{4 x} + 4 \cdot 9 e^{4 x} + e^{4 x} (2 x) + e^{4 x} \cdot 6$

Step 7.4

Combine terms.

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Step 7.4.1

Multiply $6$ by $4$ .

$4 x^{2} e^{4 x} + 24 x e^{4 x} + 4 \cdot 9 e^{4 x} + e^{4 x} (2 x) + e^{4 x} \cdot 6$

Step 7.4.2

Multiply $4$ by $9$ .

$4 x^{2} e^{4 x} + 24 x e^{4 x} + 36 e^{4 x} + e^{4 x} (2 x) + e^{4 x} \cdot 6$

Step 7.4.3

Move $6$ to the left of $e^{4 x}$ .

$4 x^{2} e^{4 x} + 24 x e^{4 x} + 36 e^{4 x} + e^{4 x} (2 x) + 6 \cdot e^{4 x}$

Step 7.4.4

Add $24 x e^{4 x}$ and $e^{4 x} (2 x)$ .

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Step 7.4.4.1

Move $e^{4 x}$ .

$4 x^{2} e^{4 x} + 24 x e^{4 x} + 2 x e^{4 x} + 36 e^{4 x} + 6 e^{4 x}$

Step 7.4.4.2

Add $24 x e^{4 x}$ and $2 x e^{4 x}$ .

$4 x^{2} e^{4 x} + 26 x e^{4 x} + 36 e^{4 x} + 6 e^{4 x}$

Step 7.4.5

Add $36 e^{4 x}$ and $6 e^{4 x}$ .

$4 x^{2} e^{4 x} + 26 x e^{4 x} + 42 e^{4 x}$

Step 7.5

Reorder terms.

$4 e^{4 x} x^{2} + 26 e^{4 x} x + 42 e^{4 x}$

Step 7.6

Reorder factors in $4 e^{4 x} x^{2} + 26 e^{4 x} x + 42 e^{4 x}$ .

$4 x^{2} e^{4 x} + 26 x e^{4 x} + 42 e^{4 x}$