Calculus Examples

$y = \cot^{-1} (\sqrt{2 t})$

Step 1

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

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\sqrt{2 t}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\sqrt{2 t}$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 t}$ as ${(2 t)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify with factoring out.

Tap for more steps...

Step 2.2.1

Factor $2$ out of $2 t$ .

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

Step 2.2.2

Apply the product rule to $2 (t)$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify the expression.

Tap for more steps...

Step 10.1

Move $2^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 10.2

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11

Multiply $2$ by $2^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 11.1

Multiply $2$ by $2^{- \frac{1}{2}}$ .

Tap for more steps...

Step 11.1.1

Raise $2$ to the power of $1$ .

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

Step 11.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11.2

Write $1$ as a fraction with a common denominator.

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

Step 11.3

Combine the numerators over the common denominator.

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

Step 11.4

Subtract $1$ from $2$ .

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

Step 12

Combine terms.

Tap for more steps...

Step 12.1

Rewrite ${\sqrt{2 t}}^{2}$ as $2 t$ .

Tap for more steps...

Step 12.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 t}$ as ${(2 t)}^{\frac{1}{2}}$ .

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

Step 12.1.2

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

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

Step 12.1.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 12.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.1.4.1

Cancel the common factor.

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

Step 12.1.4.2

Rewrite the expression.

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

Step 12.1.5

Simplify.

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

Step 12.2

Multiply $\frac{1}{2^{\frac{1}{2}} t^{\frac{1}{2}}}$ by $\frac{1}{1 + 2 t}$ .

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