Calculus Examples

$y = \frac{- 8 e^{2 x}}{7 x - 2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{- 8 e^{2 x}}{7 x - 2})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 3.1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{8 e^{2 x}}{7 x - 2}]$

Step 3.1.2

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8 e^{2 x}}{7 x - 2}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{e^{2 x}}{7 x - 2}]$ .

$- 8 \frac{d}{d x} [\frac{e^{2 x}}{7 x - 2}]$

Step 3.2

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 $f (x) = e^{2 x}$ and $g (x) = 7 x - 2$ .

$- 8 \frac{(7 x - 2) \frac{d}{d x} [e^{2 x}] - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.3

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 $g (x) = 2 x$ .

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$- 8 \frac{(7 x - 2) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]) - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.3.2

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

$- 8 \frac{(7 x - 2) (e^{u} \frac{d}{d x} [2 x]) - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.3.3

Replace all occurrences of $u$ with $2 x$ .

$- 8 \frac{(7 x - 2) (e^{2 x} \frac{d}{d x} [2 x]) - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.4

Differentiate.

Tap for more steps...

Step 3.4.1

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

$- 8 \frac{(7 x - 2) e^{2 x} (2 \frac{d}{d x} [x]) - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.4.2

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

$- 8 \frac{(7 x - 2) e^{2 x} (2 \cdot 1) - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.4.3

Simplify the expression.

Tap for more steps...

Step 3.4.3.1

Multiply $2$ by $1$ .

$- 8 \frac{(7 x - 2) e^{2 x} \cdot 2 - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.4.3.2

Move $2$ to the left of $(7 x - 2) e^{2 x}$ .

$- 8 \frac{2 \cdot ((7 x - 2) e^{2 x}) - e^{2 x} \frac{d}{d x} [7 x - 2]}{{(7 x - 2)}^{2}}$

Step 3.4.4

By the Sum Rule, the derivative of $7 x - 2$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 2]$ .

$- 8 \frac{2 (7 x - 2) e^{2 x} - e^{2 x} (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 2])}{{(7 x - 2)}^{2}}$

Step 3.4.5

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$- 8 \frac{2 (7 x - 2) e^{2 x} - e^{2 x} (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(7 x - 2)}^{2}}$

Step 3.4.6

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

$- 8 \frac{2 (7 x - 2) e^{2 x} - e^{2 x} (7 \cdot 1 + \frac{d}{d x} [- 2])}{{(7 x - 2)}^{2}}$

Step 3.4.7

Multiply $7$ by $1$ .

$- 8 \frac{2 (7 x - 2) e^{2 x} - e^{2 x} (7 + \frac{d}{d x} [- 2])}{{(7 x - 2)}^{2}}$

Step 3.4.8

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

$- 8 \frac{2 (7 x - 2) e^{2 x} - e^{2 x} (7 + 0)}{{(7 x - 2)}^{2}}$

Step 3.4.9

Combine fractions.

Tap for more steps...

Step 3.4.9.1

Add $7$ and $0$ .

$- 8 \frac{2 (7 x - 2) e^{2 x} - e^{2 x} \cdot 7}{{(7 x - 2)}^{2}}$

Step 3.4.9.2

Multiply $7$ by $- 1$ .

$- 8 \frac{2 (7 x - 2) e^{2 x} - 7 e^{2 x}}{{(7 x - 2)}^{2}}$

Step 3.4.9.3

Combine $- 8$ and $\frac{2 (7 x - 2) e^{2 x} - 7 e^{2 x}}{{(7 x - 2)}^{2}}$ .

$\frac{- 8 (2 (7 x - 2) e^{2 x} - 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.4.9.4

Move the negative in front of the fraction.

$- \frac{8 (2 (7 x - 2) e^{2 x} - 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Apply the distributive property.

$- \frac{8 ((2 (7 x) + 2 \cdot - 2) e^{2 x} - 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5.2

Apply the distributive property.

$- \frac{8 (2 (7 x) e^{2 x} + 2 \cdot - 2 e^{2 x} - 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5.3

Apply the distributive property.

$- \frac{8 (2 (7 x) e^{2 x}) + 8 (2 \cdot - 2 e^{2 x}) + 8 (- 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5.4

Simplify the numerator.

Tap for more steps...

Step 3.5.4.1

Simplify each term.

Tap for more steps...

Step 3.5.4.1.1

Multiply $7$ by $2$ .

$- \frac{8 (14 x e^{2 x}) + 8 (2 \cdot - 2 e^{2 x}) + 8 (- 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5.4.1.2

Multiply $14$ by $8$ .

$- \frac{112 (x e^{2 x}) + 8 (2 \cdot - 2 e^{2 x}) + 8 (- 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5.4.1.3

Multiply $2$ by $- 2$ .

$- \frac{112 x e^{2 x} + 8 (- 4 e^{2 x}) + 8 (- 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5.4.1.4

Multiply $- 4$ by $8$ .

$- \frac{112 x e^{2 x} - 32 e^{2 x} + 8 (- 7 e^{2 x})}{{(7 x - 2)}^{2}}$

Step 3.5.4.1.5

Multiply $- 7$ by $8$ .

$- \frac{112 x e^{2 x} - 32 e^{2 x} - 56 e^{2 x}}{{(7 x - 2)}^{2}}$

Step 3.5.4.2

Subtract $56 e^{2 x}$ from $- 32 e^{2 x}$ .

$- \frac{112 x e^{2 x} - 88 e^{2 x}}{{(7 x - 2)}^{2}}$

Step 3.5.5

Factor $8 e^{2 x}$ out of $112 x e^{2 x} - 88 e^{2 x}$ .

Tap for more steps...

Step 3.5.5.1

Factor $8 e^{2 x}$ out of $112 x e^{2 x}$ .

$- \frac{8 e^{2 x} (14 x) - 88 e^{2 x}}{{(7 x - 2)}^{2}}$

Step 3.5.5.2

Factor $8 e^{2 x}$ out of $- 88 e^{2 x}$ .

$- \frac{8 e^{2 x} (14 x) + 8 e^{2 x} (- 11)}{{(7 x - 2)}^{2}}$

Step 3.5.5.3

Factor $8 e^{2 x}$ out of $8 e^{2 x} (14 x) + 8 e^{2 x} (- 11)$ .

$- \frac{8 e^{2 x} (14 x - 11)}{{(7 x - 2)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{8 e^{2 x} (14 x - 11)}{{(7 x - 2)}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{8 e^{2 x} (14 x - 11)}{{(7 x - 2)}^{2}}$