Calculus Examples

$t \sqrt{4 - t}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - t}$ as ${(4 - t)}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t$ and $g (t) = {(4 - t)}^{\frac{1}{2}}$ .

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

Step 3

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 $g (t) = 4 - t$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $4 - t$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $4 - t$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

Combine $\frac{1}{2}$ and ${(4 - t)}^{- \frac{1}{2}}$ .

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

Step 8.3

Move ${(4 - t)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.4

Combine $\frac{1}{2 {(4 - t)}^{\frac{1}{2}}}$ and $t$ .

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

Step 9

By the Sum Rule, the derivative of $4 - t$ with respect to $t$ is $\frac{d}{d t} [4] + \frac{d}{d t} [- t]$ .

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

Step 10

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$\frac{t}{2 {(4 - t)}^{\frac{1}{2}}} (0 + \frac{d}{d t} [- t]) + {(4 - t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Combine fractions.

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Step 14.1

Multiply $- 1$ by $1$ .

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

Step 14.2

Combine $\frac{t}{2 {(4 - t)}^{\frac{1}{2}}}$ and $- 1$ .

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

Step 14.3

Simplify the expression.

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Step 14.3.1

Move $- 1$ to the left of $t$ .

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

Step 14.3.2

Rewrite $- 1 t$ as $- t$ .

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

Step 14.3.3

Move the negative in front of the fraction.

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

Step 15

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

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

Step 16

Multiply ${(4 - t)}^{\frac{1}{2}}$ by $1$ .

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

Step 17

To write ${(4 - t)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(4 - t)}^{\frac{1}{2}}}{2 {(4 - t)}^{\frac{1}{2}}}$ .

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

Step 18

Combine ${(4 - t)}^{\frac{1}{2}}$ and $\frac{2 {(4 - t)}^{\frac{1}{2}}}{2 {(4 - t)}^{\frac{1}{2}}}$ .

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

Step 19

Combine the numerators over the common denominator.

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

Step 20

Multiply ${(4 - t)}^{\frac{1}{2}}$ by ${(4 - t)}^{\frac{1}{2}}$ by adding the exponents.

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Step 20.1

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

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

Step 20.2

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

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

Step 20.3

Combine the numerators over the common denominator.

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

Step 20.4

Add $1$ and $1$ .

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

Step 20.5

Divide $2$ by $2$ .

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

Step 21

Simplify ${(4 - t)}^{1} \cdot 2$ .

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

Step 22

Move $2$ to the left of $4 - t$ .

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

Step 23

Simplify.

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Step 23.1

Apply the distributive property.

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

Step 23.2

Simplify the numerator.

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Step 23.2.1

Simplify each term.

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Step 23.2.1.1

Multiply $2$ by $4$ .

$\frac{- t + 8 + 2 (- t)}{2 {(4 - t)}^{\frac{1}{2}}}$

Step 23.2.1.2

Multiply $- 1$ by $2$ .

$\frac{- t + 8 - 2 t}{2 {(4 - t)}^{\frac{1}{2}}}$

Step 23.2.2

Subtract $2 t$ from $- t$ .

$\frac{- 3 t + 8}{2 {(4 - t)}^{\frac{1}{2}}}$

Step 23.3

Factor $- 1$ out of $- 3 t$ .

$\frac{- (3 t) + 8}{2 {(4 - t)}^{\frac{1}{2}}}$

Step 23.4

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{- (3 t) - 1 (- 8)}{2 {(4 - t)}^{\frac{1}{2}}}$

Step 23.5

Factor $- 1$ out of $- (3 t) - 1 (- 8)$ .

$\frac{- (3 t - 8)}{2 {(4 - t)}^{\frac{1}{2}}}$

Step 23.6

Rewrite $- (3 t - 8)$ as $- 1 (3 t - 8)$ .

$\frac{- 1 (3 t - 8)}{2 {(4 - t)}^{\frac{1}{2}}}$

Step 23.7

Move the negative in front of the fraction.

$- \frac{3 t - 8}{2 {(4 - t)}^{\frac{1}{2}}}$