Calculus Examples

$\frac{8 e^{6 x}}{5 x + 2}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 e^{6 x}}{5 x + 2}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{e^{6 x}}{5 x + 2}]$ .

$8 \frac{d}{d x} [\frac{e^{6 x}}{5 x + 2}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{6 x}$ and $g (x) = 5 x + 2$ .

$8 \frac{(5 x + 2) \frac{d}{d x} [e^{6 x}] - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

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 $g (x) = 6 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $6 x$ .

$8 \frac{(5 x + 2) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 3.2

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

$8 \frac{(5 x + 2) (e^{u} \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 3.3

Replace all occurrences of $u$ with $6 x$ .

$8 \frac{(5 x + 2) (e^{6 x} \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$8 \frac{(5 x + 2) e^{6 x} (6 \frac{d}{d x} [x]) - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4.2

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

$8 \frac{(5 x + 2) e^{6 x} (6 \cdot 1) - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $6$ by $1$ .

$8 \frac{(5 x + 2) e^{6 x} \cdot 6 - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4.3.2

Move $6$ to the left of $(5 x + 2) e^{6 x}$ .

$8 \frac{6 \cdot ((5 x + 2) e^{6 x}) - e^{6 x} \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4.4

By the Sum Rule, the derivative of $5 x + 2$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [2]$ .

$8 \frac{6 (5 x + 2) e^{6 x} - e^{6 x} (\frac{d}{d x} [5 x] + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 4.5

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

$8 \frac{6 (5 x + 2) e^{6 x} - e^{6 x} (5 \frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 4.6

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

$8 \frac{6 (5 x + 2) e^{6 x} - e^{6 x} (5 \cdot 1 + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 4.7

Multiply $5$ by $1$ .

$8 \frac{6 (5 x + 2) e^{6 x} - e^{6 x} (5 + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 4.8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$8 \frac{6 (5 x + 2) e^{6 x} - e^{6 x} (5 + 0)}{{(5 x + 2)}^{2}}$

Step 4.9

Combine fractions.

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Step 4.9.1

Add $5$ and $0$ .

$8 \frac{6 (5 x + 2) e^{6 x} - e^{6 x} \cdot 5}{{(5 x + 2)}^{2}}$

Step 4.9.2

Multiply $5$ by $- 1$ .

$8 \frac{6 (5 x + 2) e^{6 x} - 5 e^{6 x}}{{(5 x + 2)}^{2}}$

Step 4.9.3

Combine $8$ and $\frac{6 (5 x + 2) e^{6 x} - 5 e^{6 x}}{{(5 x + 2)}^{2}}$ .

$\frac{8 (6 (5 x + 2) e^{6 x} - 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{8 ((6 (5 x) + 6 \cdot 2) e^{6 x} - 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{8 (6 (5 x) e^{6 x} + 6 \cdot 2 e^{6 x} - 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5.3

Apply the distributive property.

$\frac{8 (6 (5 x) e^{6 x}) + 8 (6 \cdot 2 e^{6 x}) + 8 (- 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply $5$ by $6$ .

$\frac{8 (30 x e^{6 x}) + 8 (6 \cdot 2 e^{6 x}) + 8 (- 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5.4.1.2

Multiply $30$ by $8$ .

$\frac{240 (x e^{6 x}) + 8 (6 \cdot 2 e^{6 x}) + 8 (- 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5.4.1.3

Multiply $6$ by $2$ .

$\frac{240 x e^{6 x} + 8 (12 e^{6 x}) + 8 (- 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5.4.1.4

Multiply $12$ by $8$ .

$\frac{240 x e^{6 x} + 96 e^{6 x} + 8 (- 5 e^{6 x})}{{(5 x + 2)}^{2}}$

Step 5.4.1.5

Multiply $- 5$ by $8$ .

$\frac{240 x e^{6 x} + 96 e^{6 x} - 40 e^{6 x}}{{(5 x + 2)}^{2}}$

Step 5.4.2

Subtract $40 e^{6 x}$ from $96 e^{6 x}$ .

$\frac{240 x e^{6 x} + 56 e^{6 x}}{{(5 x + 2)}^{2}}$

Step 5.5

Reorder terms.

$\frac{240 e^{6 x} x + 56 e^{6 x}}{{(5 x + 2)}^{2}}$

Step 5.6

Factor $8 e^{6 x}$ out of $240 e^{6 x} x + 56 e^{6 x}$ .

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Step 5.6.1

Factor $8 e^{6 x}$ out of $240 e^{6 x} x$ .

$\frac{8 e^{6 x} (30 x) + 56 e^{6 x}}{{(5 x + 2)}^{2}}$

Step 5.6.2

Factor $8 e^{6 x}$ out of $56 e^{6 x}$ .

$\frac{8 e^{6 x} (30 x) + 8 e^{6 x} (7)}{{(5 x + 2)}^{2}}$

Step 5.6.3

Factor $8 e^{6 x}$ out of $8 e^{6 x} (30 x) + 8 e^{6 x} (7)$ .

$\frac{8 e^{6 x} (30 x + 7)}{{(5 x + 2)}^{2}}$