Calculus Examples

$f (t) = arccsc (- 4 t^{2})$

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

The derivative of $arccsc (u)$ with respect to $u$ is $- \frac{1}{u \sqrt{u^{2} - 1}}$ .

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

Step 1.3

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

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Factor $- 4$ out of $- 4 t^{2}$ .

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

Step 2.2

Simplify the expression.

Tap for more steps...

Step 2.2.1

Apply the product rule to $- 4 (t^{2})$ .

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

Step 2.2.2

Raise $- 4$ to the power of $2$ .

$- \frac{1}{- 4 t^{2} \sqrt{16 {(t^{2})}^{2} - 1}} \frac{d}{d t} [- 4 t^{2}]$

Step 2.2.3

Multiply the exponents in ${(t^{2})}^{2}$ .

Tap for more steps...

Step 2.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{- 4 t^{2} \sqrt{16 t^{2 \cdot 2} - 1}} \frac{d}{d t} [- 4 t^{2}]$

Step 2.2.3.2

Multiply $2$ by $2$ .

$- \frac{1}{- 4 t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [- 4 t^{2}]$

Step 2.2.4

Move the negative in front of the fraction.

$- - \frac{1}{4 t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [- 4 t^{2}]$

Step 2.2.5

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{4 t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [- 4 t^{2}]$

Step 2.2.6

Multiply $\frac{1}{4 t^{2} \sqrt{16 t^{4} - 1}}$ by $1$ .

$\frac{1}{4 t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [- 4 t^{2}]$

Step 2.3

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4 t^{2}$ with respect to $t$ is $- 4 \frac{d}{d t} [t^{2}]$ .

$\frac{1}{4 t^{2} \sqrt{16 t^{4} - 1}} (- 4 \frac{d}{d t} [t^{2}])$

Step 2.4

Simplify terms.

Tap for more steps...

Step 2.4.1

Combine $- 4$ and $\frac{1}{4 t^{2} \sqrt{16 t^{4} - 1}}$ .

$\frac{- 4}{4 t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [t^{2}]$

Step 2.4.2

Cancel the common factor of $- 4$ and $4$ .

Tap for more steps...

Step 2.4.2.1

Factor $4$ out of $- 4$ .

$\frac{4 \cdot - 1}{4 t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [t^{2}]$

Step 2.4.2.2

Cancel the common factors.

Tap for more steps...

Step 2.4.2.2.1

Factor $4$ out of $4 t^{2} \sqrt{16 t^{4} - 1}$ .

$\frac{4 \cdot - 1}{4 (t^{2} \sqrt{16 t^{4} - 1})} \frac{d}{d t} [t^{2}]$

Step 2.4.2.2.2

Cancel the common factor.

$\frac{4 \cdot - 1}{4 (t^{2} \sqrt{16 t^{4} - 1})} \frac{d}{d t} [t^{2}]$

Step 2.4.2.2.3

Rewrite the expression.

$\frac{- 1}{t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [t^{2}]$

Step 2.4.3

Move the negative in front of the fraction.

$- \frac{1}{t^{2} \sqrt{16 t^{4} - 1}} \frac{d}{d t} [t^{2}]$

Step 2.5

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

$- \frac{1}{t^{2} \sqrt{16 t^{4} - 1}} (2 t)$

Step 2.6

Simplify terms.

Tap for more steps...

Step 2.6.1

Multiply $2$ by $- 1$ .

$- 2 \frac{1}{t^{2} \sqrt{16 t^{4} - 1}} t$

Step 2.6.2

Combine $- 2$ and $\frac{1}{t^{2} \sqrt{16 t^{4} - 1}}$ .

$\frac{- 2}{t^{2} \sqrt{16 t^{4} - 1}} t$

Step 2.6.3

Combine $\frac{- 2}{t^{2} \sqrt{16 t^{4} - 1}}$ and $t$ .

$\frac{- 2 t}{t^{2} \sqrt{16 t^{4} - 1}}$

Step 2.6.4

Cancel the common factor of $t$ and $t^{2}$ .

Tap for more steps...

Step 2.6.4.1

Factor $t$ out of $- 2 t$ .

$\frac{t \cdot - 2}{t^{2} \sqrt{16 t^{4} - 1}}$

Step 2.6.4.2

Cancel the common factors.

Tap for more steps...

Step 2.6.4.2.1

Factor $t$ out of $t^{2} \sqrt{16 t^{4} - 1}$ .

$\frac{t \cdot - 2}{t (t \sqrt{16 t^{4} - 1})}$

Step 2.6.4.2.2

Cancel the common factor.

$\frac{t \cdot - 2}{t (t \sqrt{16 t^{4} - 1})}$

Step 2.6.4.2.3

Rewrite the expression.

$\frac{- 2}{t \sqrt{16 t^{4} - 1}}$

Step 2.6.5

Move the negative in front of the fraction.

$- \frac{2}{t \sqrt{16 t^{4} - 1}}$