Calculus Examples

$\frac{800 t}{t + 2}$

Step 1

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 $f (t) = 800 t$ and $g (t) = t + 2$ .

$\frac{(t + 2) \frac{d}{d t} [800 t] - 1 \cdot 800 t \frac{d}{d t} [t + 2]}{{(t + 2)}^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Since $800$ is constant with respect to $t$ , the derivative of $800 t$ with respect to $t$ is $800 \frac{d}{d t} [t]$ .

$\frac{(t + 2) (800 \frac{d}{d t} [t]) - 1 \cdot 800 t \frac{d}{d t} [t + 2]}{{(t + 2)}^{2}}$

Step 2.2

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

$\frac{(t + 2) (800 \cdot 1) - 1 \cdot 800 t \frac{d}{d t} [t + 2]}{{(t + 2)}^{2}}$

Step 2.3

Multiply $800$ by $1$ .

$\frac{(t + 2) \cdot 800 - 1 \cdot 800 t \frac{d}{d t} [t + 2]}{{(t + 2)}^{2}}$

Step 2.4

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

$\frac{(t + 2) \cdot 800 - 1 \cdot 800 t (\frac{d}{d t} [t] + \frac{d}{d t} [2])}{{(t + 2)}^{2}}$

Step 2.5

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

$\frac{(t + 2) \cdot 800 - 1 \cdot 800 t (1 + \frac{d}{d t} [2])}{{(t + 2)}^{2}}$

Step 2.6

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

$\frac{(t + 2) \cdot 800 - 1 \cdot 800 t (1 + 0)}{{(t + 2)}^{2}}$

Step 2.7

Add $1$ and $0$ .

$\frac{(t + 2) \cdot 800 - 1 \cdot 800 t \cdot 1}{{(t + 2)}^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{t \cdot 800 + 2 \cdot 800 - 1 \cdot 800 t \cdot 1}{{(t + 2)}^{2}}$

Step 3.2

Simplify the numerator.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Move $800$ to the left of $t$ .

$\frac{800 \cdot t + 2 \cdot 800 - 1 \cdot 800 t \cdot 1}{{(t + 2)}^{2}}$

Step 3.2.1.2

Multiply $2$ by $800$ .

$\frac{800 t + 1600 - 1 \cdot 800 t \cdot 1}{{(t + 2)}^{2}}$

Step 3.2.1.3

Multiply $- 1$ by $800$ .

$\frac{800 t + 1600 - 800 t \cdot 1}{{(t + 2)}^{2}}$

Step 3.2.1.4

Multiply $- 800$ by $1$ .

$\frac{800 t + 1600 - 800 t}{{(t + 2)}^{2}}$

Step 3.2.2

Combine the opposite terms in $800 t + 1600 - 800 t$ .

Tap for more steps...

Step 3.2.2.1

Subtract $800 t$ from $800 t$ .

$\frac{0 + 1600}{{(t + 2)}^{2}}$

Step 3.2.2.2

Add $0$ and $1600$ .

$\frac{1600}{{(t + 2)}^{2}}$