Calculus Examples

$\sqrt{1 - t^{2}}$

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $1 - t^{2}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 - t^{2}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

By the Sum Rule, the derivative of $1 - t^{2}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- t^{2}]$ .

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

Step 9

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$\frac{1}{2 {(1 - t^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d t} [- t^{2}])$

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify terms.

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Step 13.1

Multiply $2$ by $- 1$ .

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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

Step 14

Cancel the common factors.

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Step 14.1

Factor $2$ out of $2 {(1 - t^{2})}^{\frac{1}{2}}$ .

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

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

Step 15

Move the negative in front of the fraction.

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