Calculus Examples

$g (x) = \frac{e^{x}}{1 - 4 x}$

Step 1

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 $f (x) = e^{x}$ and $g (x) = 1 - 4 x$ .

$\frac{(1 - 4 x) \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [1 - 4 x]}{{(1 - 4 x)}^{2}}$

Step 2

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

$\frac{(1 - 4 x) e^{x} - e^{x} \frac{d}{d x} [1 - 4 x]}{{(1 - 4 x)}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - 4 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x]$ .

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

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(1 - 4 x) e^{x} - e^{x} (0 + \frac{d}{d x} [- 4 x])}{{(1 - 4 x)}^{2}}$

Step 3.3

Add $0$ and $\frac{d}{d x} [- 4 x]$ .

$\frac{(1 - 4 x) e^{x} - e^{x} \frac{d}{d x} [- 4 x]}{{(1 - 4 x)}^{2}}$

Step 3.4

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

$\frac{(1 - 4 x) e^{x} - e^{x} (- 4 \frac{d}{d x} [x])}{{(1 - 4 x)}^{2}}$

Step 3.5

Multiply $- 4$ by $- 1$ .

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

Step 3.6

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

$\frac{(1 - 4 x) e^{x} + 4 e^{x} \cdot 1}{{(1 - 4 x)}^{2}}$

Step 3.7

Multiply $4$ by $1$ .

$\frac{(1 - 4 x) e^{x} + 4 e^{x}}{{(1 - 4 x)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{1 e^{x} - 4 x e^{x} + 4 e^{x}}{{(1 - 4 x)}^{2}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Multiply $e^{x}$ by $1$ .

$\frac{e^{x} - 4 x e^{x} + 4 e^{x}}{{(1 - 4 x)}^{2}}$

Step 4.2.2

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

$\frac{- 4 x e^{x} + 5 e^{x}}{{(1 - 4 x)}^{2}}$

Step 4.3

Reorder terms.

$\frac{- 4 e^{x} x + 5 e^{x}}{{(- 4 x + 1)}^{2}}$

Step 4.4

Factor $e^{x}$ out of $- 4 e^{x} x + 5 e^{x}$ .

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Step 4.4.1

Factor $e^{x}$ out of $- 4 e^{x} x$ .

$\frac{e^{x} (- 4 x) + 5 e^{x}}{{(- 4 x + 1)}^{2}}$

Step 4.4.2

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

$\frac{e^{x} (- 4 x) + e^{x} \cdot 5}{{(- 4 x + 1)}^{2}}$

Step 4.4.3

Factor $e^{x}$ out of $e^{x} (- 4 x) + e^{x} \cdot 5$ .

$\frac{e^{x} (- 4 x + 5)}{{(- 4 x + 1)}^{2}}$

Step 4.5

Factor $- 1$ out of $- 4 x$ .

$\frac{e^{x} (- (4 x) + 5)}{{(- 4 x + 1)}^{2}}$

Step 4.6

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{e^{x} (- (4 x) - 1 (- 5))}{{(- 4 x + 1)}^{2}}$

Step 4.7

Factor $- 1$ out of $- (4 x) - 1 (- 5)$ .

$\frac{e^{x} (- (4 x - 5))}{{(- 4 x + 1)}^{2}}$

Step 4.8

Rewrite $- (4 x - 5)$ as $- 1 (4 x - 5)$ .

$\frac{e^{x} (- 1 (4 x - 5))}{{(- 4 x + 1)}^{2}}$

Step 4.9

Move the negative in front of the fraction.

$- \frac{e^{x} (4 x - 5)}{{(- 4 x + 1)}^{2}}$