Calculus Examples

$y = \frac{- 8 e^{5 x}}{3 x - 2}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{8 e^{5 x}}{3 x - 2}]$

Step 1.2

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8 e^{5 x}}{3 x - 2}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{e^{5 x}}{3 x - 2}]$ .

$- 8 \frac{d}{d x} [\frac{e^{5 x}}{3 x - 2}]$

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

$- 8 \frac{(3 x - 2) \frac{d}{d x} [e^{5 x}] - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

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 $g (x) = 5 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $5 x$ .

$- 8 \frac{(3 x - 2) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x]) - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 3.2

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

$- 8 \frac{(3 x - 2) (e^{u} \frac{d}{d x} [5 x]) - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 3.3

Replace all occurrences of $u$ with $5 x$ .

$- 8 \frac{(3 x - 2) (e^{5 x} \frac{d}{d x} [5 x]) - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$- 8 \frac{(3 x - 2) e^{5 x} (5 \frac{d}{d x} [x]) - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4.2

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

$- 8 \frac{(3 x - 2) e^{5 x} (5 \cdot 1) - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $5$ by $1$ .

$- 8 \frac{(3 x - 2) e^{5 x} \cdot 5 - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4.3.2

Move $5$ to the left of $(3 x - 2) e^{5 x}$ .

$- 8 \frac{5 \cdot ((3 x - 2) e^{5 x}) - e^{5 x} \frac{d}{d x} [3 x - 2]}{{(3 x - 2)}^{2}}$

Step 4.4

By the Sum Rule, the derivative of $3 x - 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$ .

$- 8 \frac{5 (3 x - 2) e^{5 x} - e^{5 x} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.5

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

$- 8 \frac{5 (3 x - 2) e^{5 x} - e^{5 x} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.6

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

$- 8 \frac{5 (3 x - 2) e^{5 x} - e^{5 x} (3 \cdot 1 + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.7

Multiply $3$ by $1$ .

$- 8 \frac{5 (3 x - 2) e^{5 x} - e^{5 x} (3 + \frac{d}{d x} [- 2])}{{(3 x - 2)}^{2}}$

Step 4.8

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

$- 8 \frac{5 (3 x - 2) e^{5 x} - e^{5 x} (3 + 0)}{{(3 x - 2)}^{2}}$

Step 4.9

Combine fractions.

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Step 4.9.1

Add $3$ and $0$ .

$- 8 \frac{5 (3 x - 2) e^{5 x} - e^{5 x} \cdot 3}{{(3 x - 2)}^{2}}$

Step 4.9.2

Multiply $3$ by $- 1$ .

$- 8 \frac{5 (3 x - 2) e^{5 x} - 3 e^{5 x}}{{(3 x - 2)}^{2}}$

Step 4.9.3

Combine $- 8$ and $\frac{5 (3 x - 2) e^{5 x} - 3 e^{5 x}}{{(3 x - 2)}^{2}}$ .

$\frac{- 8 (5 (3 x - 2) e^{5 x} - 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 4.9.4

Move the negative in front of the fraction.

$- \frac{8 (5 (3 x - 2) e^{5 x} - 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- \frac{8 ((5 (3 x) + 5 \cdot - 2) e^{5 x} - 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 5.2

Apply the distributive property.

$- \frac{8 (5 (3 x) e^{5 x} + 5 \cdot - 2 e^{5 x} - 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 5.3

Apply the distributive property.

$- \frac{8 (5 (3 x) e^{5 x}) + 8 (5 \cdot - 2 e^{5 x}) + 8 (- 3 e^{5 x})}{{(3 x - 2)}^{2}}$

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 $3$ by $5$ .

$- \frac{8 (15 x e^{5 x}) + 8 (5 \cdot - 2 e^{5 x}) + 8 (- 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 5.4.1.2

Multiply $15$ by $8$ .

$- \frac{120 (x e^{5 x}) + 8 (5 \cdot - 2 e^{5 x}) + 8 (- 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 5.4.1.3

Multiply $5$ by $- 2$ .

$- \frac{120 x e^{5 x} + 8 (- 10 e^{5 x}) + 8 (- 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 5.4.1.4

Multiply $- 10$ by $8$ .

$- \frac{120 x e^{5 x} - 80 e^{5 x} + 8 (- 3 e^{5 x})}{{(3 x - 2)}^{2}}$

Step 5.4.1.5

Multiply $- 3$ by $8$ .

$- \frac{120 x e^{5 x} - 80 e^{5 x} - 24 e^{5 x}}{{(3 x - 2)}^{2}}$

Step 5.4.2

Subtract $24 e^{5 x}$ from $- 80 e^{5 x}$ .

$- \frac{120 x e^{5 x} - 104 e^{5 x}}{{(3 x - 2)}^{2}}$

Step 5.5

Factor $8 e^{5 x}$ out of $120 x e^{5 x} - 104 e^{5 x}$ .

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Step 5.5.1

Factor $8 e^{5 x}$ out of $120 x e^{5 x}$ .

$- \frac{8 e^{5 x} (15 x) - 104 e^{5 x}}{{(3 x - 2)}^{2}}$

Step 5.5.2

Factor $8 e^{5 x}$ out of $- 104 e^{5 x}$ .

$- \frac{8 e^{5 x} (15 x) + 8 e^{5 x} (- 13)}{{(3 x - 2)}^{2}}$

Step 5.5.3

Factor $8 e^{5 x}$ out of $8 e^{5 x} (15 x) + 8 e^{5 x} (- 13)$ .

$- \frac{8 e^{5 x} (15 x - 13)}{{(3 x - 2)}^{2}}$