Calculus Examples

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

Step 1

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

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

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 $g (t) = \frac{t}{t^{2} + 1}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\frac{t}{t^{2} + 1}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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 $f (t) = t$ and $g (t) = t^{2} + 1$ .

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

Step 9

Differentiate.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

Simplify the expression.

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Step 9.6.1

Add $2 t$ and $0$ .

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

Step 9.6.2

Multiply $2$ by $- 1$ .

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Add $1$ and $1$ .

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

Step 14

Subtract $2 t^{2}$ from $t^{2}$ .

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

Step 15

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

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

Step 16

Move $2$ to the left of ${(t^{2} + 1)}^{2}$ .

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

Step 17

Simplify.

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Step 17.1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 17.2

Apply the product rule to $\frac{t^{2} + 1}{t}$ .

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

Step 17.3

Combine terms.

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Step 17.3.1

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

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

Step 17.3.2

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

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

Step 17.3.3

Multiply ${(t^{2} + 1)}^{2}$ by ${(t^{2} + 1)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 17.3.3.1

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

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

Step 17.3.3.2

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

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

Step 17.3.3.3

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

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

Step 17.3.3.4

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

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

Step 17.3.3.5

Combine the numerators over the common denominator.

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

Step 17.3.3.6

Simplify the numerator.

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Step 17.3.3.6.1

Multiply $2$ by $2$ .

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

Step 17.3.3.6.2

Add $- 1$ and $4$ .

$\frac{- t^{2} + 1}{2 {(t^{2} + 1)}^{\frac{3}{2}} t^{\frac{1}{2}}}$

Step 17.4

Reorder terms.

$\frac{- t^{2} + 1}{2 t^{\frac{1}{2}} {(t^{2} + 1)}^{\frac{3}{2}}}$

Step 17.5

Simplify the numerator.

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Step 17.5.1

Rewrite $1$ as $1^{2}$ .

$\frac{- t^{2} + 1^{2}}{2 t^{\frac{1}{2}} {(t^{2} + 1)}^{\frac{3}{2}}}$

Step 17.5.2

Reorder $- t^{2}$ and $1^{2}$ .

$\frac{1^{2} - t^{2}}{2 t^{\frac{1}{2}} {(t^{2} + 1)}^{\frac{3}{2}}}$

Step 17.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = t$ .

$\frac{(1 + t) (1 - t)}{2 t^{\frac{1}{2}} {(t^{2} + 1)}^{\frac{3}{2}}}$