Calculus Examples

$\frac{180 t}{t + 4}$

Step 1

Since $180$ is constant with respect to $t$ , the derivative of $\frac{180 t}{t + 4}$ with respect to $t$ is $180 \frac{d}{d t} [\frac{t}{t + 4}]$ .

$180 \frac{d}{d t} [\frac{t}{t + 4}]$

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 $g (t) = t + 4$ .

$180 \frac{(t + 4) \frac{d}{d t} [t] - t \frac{d}{d t} [t + 4]}{{(t + 4)}^{2}}$

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 $n = 1$ .

$180 \frac{(t + 4) \cdot 1 - t \frac{d}{d t} [t + 4]}{{(t + 4)}^{2}}$

Step 3.2

Multiply $t + 4$ by $1$ .

$180 \frac{t + 4 - t \frac{d}{d t} [t + 4]}{{(t + 4)}^{2}}$

Step 3.3

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

$180 \frac{t + 4 - t (\frac{d}{d t} [t] + \frac{d}{d t} [4])}{{(t + 4)}^{2}}$

Step 3.4

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

$180 \frac{t + 4 - t (1 + \frac{d}{d t} [4])}{{(t + 4)}^{2}}$

Step 3.5

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$180 \frac{t + 4 - t (1 + 0)}{{(t + 4)}^{2}}$

Step 3.6

Simplify terms.

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Step 3.6.1

Add $1$ and $0$ .

$180 \frac{t + 4 - t \cdot 1}{{(t + 4)}^{2}}$

Step 3.6.2

Multiply $- 1$ by $1$ .

$180 \frac{t + 4 - t}{{(t + 4)}^{2}}$

Step 3.6.3

Subtract $t$ from $t$ .

$180 \frac{0 + 4}{{(t + 4)}^{2}}$

Step 3.6.4

Add $0$ and $4$ .

$180 \frac{4}{{(t + 4)}^{2}}$

Step 3.6.5

Combine $180$ and $\frac{4}{{(t + 4)}^{2}}$ .

$\frac{180 \cdot 4}{{(t + 4)}^{2}}$

Step 3.6.6

Multiply $180$ by $4$ .

$\frac{720}{{(t + 4)}^{2}}$