Calculus Examples

$p (t) = \frac{2000 t}{4 t + 75}$

Step 1

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

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t$ and $g (t) = 4 t + 75$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $4$ by $1$ .

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

Step 3.7

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

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

Step 3.8

Simplify terms.

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Step 3.8.1

Add $4$ and $0$ .

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

Step 3.8.2

Multiply $4$ by $- 1$ .

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

Step 3.8.3

Subtract $4 t$ from $4 t$ .

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

Step 3.8.4

Add $0$ and $75$ .

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

Step 3.8.5

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

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

Step 3.8.6

Multiply $2000$ by $75$ .

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