Calculus Examples

$\frac{1}{4 - t^{2}}$

Step 1

Rewrite $\frac{1}{4 - t^{2}}$ as ${(4 - t^{2})}^{- 1}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = 4 - t^{2}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $4 - t^{2}$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [4 - t^{2}]$

Step 2.2

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

$- u^{- 2} \frac{d}{d t} [4 - t^{2}]$

Step 2.3

Replace all occurrences of $u$ with $4 - t^{2}$ .

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $4 - t^{2}$ with respect to $t$ is $\frac{d}{d t} [4] + \frac{d}{d t} [- t^{2}]$ .

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

Step 3.2

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

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

Step 3.3

Add $0$ and $\frac{d}{d t} [- t^{2}]$ .

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

Step 3.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{2}$ with respect to $t$ is $- \frac{d}{d t} [t^{2}]$ .

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

Step 3.5

Multiply.

Tap for more steps...

Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

Multiply ${(4 - t^{2})}^{- 2}$ by $1$ .

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

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

${(4 - t^{2})}^{- 2} (2 t)$

Step 3.7

Move $2$ to the left of ${(4 - t^{2})}^{- 2}$ .

$2 \cdot {(4 - t^{2})}^{- 2} t$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine terms.

Tap for more steps...

Step 5.1.1

Combine $2$ and $\frac{1}{{(4 - t^{2})}^{2}}$ .

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

Step 5.1.2

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

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

Step 5.2

Reorder terms.

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