Calculus Examples

$y = \sqrt{\frac{t}{t + 1}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} ({(\frac{t}{t + 1})}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $t$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

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

Step 4.5.2

Subtract $2$ from $1$ .

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

Step 4.6

Move the negative in front of the fraction.

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

Step 4.7

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 $g (t) = t + 1$ .

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

Step 4.8

Differentiate.

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Step 4.8.1

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

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

Step 4.8.2

Multiply $t + 1$ by $1$ .

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

Step 4.8.3

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

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

Step 4.8.4

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

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

Step 4.8.5

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

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

Step 4.8.6

Simplify terms.

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Step 4.8.6.1

Add $1$ and $0$ .

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

Step 4.8.6.2

Multiply $- 1$ by $1$ .

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

Step 4.8.6.3

Subtract $t$ from $t$ .

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

Step 4.8.6.4

Add $0$ and $1$ .

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

Step 4.8.6.5

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

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

Step 4.8.6.6

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

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

Step 4.9

Simplify.

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Step 4.9.1

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

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

Step 4.9.2

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

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

Step 4.9.3

Combine terms.

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Step 4.9.3.1

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

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

Step 4.9.3.2

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

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

Step 4.9.3.3

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

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Step 4.9.3.3.1

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

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

Step 4.9.3.3.2

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

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

Step 4.9.3.3.3

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

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

Step 4.9.3.3.4

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

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

Step 4.9.3.3.5

Combine the numerators over the common denominator.

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

Step 4.9.3.3.6

Simplify the numerator.

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Step 4.9.3.3.6.1

Multiply $2$ by $2$ .

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

Step 4.9.3.3.6.2

Add $- 1$ and $4$ .

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

Step 4.9.4

Reorder terms.

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{1}{2 t^{\frac{1}{2}} {(t + 1)}^{\frac{3}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d t}$ .

$\frac{d y}{d t} = \frac{1}{2 t^{\frac{1}{2}} {(t + 1)}^{\frac{3}{2}}}$