Calculus Examples

$\frac{2000 x}{4 x + 75}$

Step 1

Since $2000$ is constant with respect to $x$ , the derivative of $\frac{2000 x}{4 x + 75}$ with respect to $x$ is $2000 \frac{d}{d x} [\frac{x}{4 x + 75}]$ .

$2000 \frac{d}{d x} [\frac{x}{4 x + 75}]$

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 $f (x) = x$ and $g (x) = 4 x + 75$ .

$2000 \frac{(4 x + 75) \frac{d}{d x} [x] - x \frac{d}{d x} [4 x + 75]}{{(4 x + 75)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$2000 \frac{(4 x + 75) \cdot 1 - x \frac{d}{d x} [4 x + 75]}{{(4 x + 75)}^{2}}$

Step 3.2

Multiply $4 x + 75$ by $1$ .

$2000 \frac{4 x + 75 - x \frac{d}{d x} [4 x + 75]}{{(4 x + 75)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $4 x + 75$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [75]$ .

$2000 \frac{4 x + 75 - x (\frac{d}{d x} [4 x] + \frac{d}{d x} [75])}{{(4 x + 75)}^{2}}$

Step 3.4

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

$2000 \frac{4 x + 75 - x (4 \frac{d}{d x} [x] + \frac{d}{d x} [75])}{{(4 x + 75)}^{2}}$

Step 3.5

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

$2000 \frac{4 x + 75 - x (4 \cdot 1 + \frac{d}{d x} [75])}{{(4 x + 75)}^{2}}$

Step 3.6

Multiply $4$ by $1$ .

$2000 \frac{4 x + 75 - x (4 + \frac{d}{d x} [75])}{{(4 x + 75)}^{2}}$

Step 3.7

Since $75$ is constant with respect to $x$ , the derivative of $75$ with respect to $x$ is $0$ .

$2000 \frac{4 x + 75 - x (4 + 0)}{{(4 x + 75)}^{2}}$

Step 3.8

Simplify terms.

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Step 3.8.1

Add $4$ and $0$ .

$2000 \frac{4 x + 75 - x \cdot 4}{{(4 x + 75)}^{2}}$

Step 3.8.2

Multiply $4$ by $- 1$ .

$2000 \frac{4 x + 75 - 4 x}{{(4 x + 75)}^{2}}$

Step 3.8.3

Subtract $4 x$ from $4 x$ .

$2000 \frac{0 + 75}{{(4 x + 75)}^{2}}$

Step 3.8.4

Add $0$ and $75$ .

$2000 \frac{75}{{(4 x + 75)}^{2}}$

Step 3.8.5

Combine $2000$ and $\frac{75}{{(4 x + 75)}^{2}}$ .

$\frac{2000 \cdot 75}{{(4 x + 75)}^{2}}$

Step 3.8.6

Multiply $2000$ by $75$ .

$\frac{150000}{{(4 x + 75)}^{2}}$