Calculus Examples

$f (x) = \frac{4 t}{3 \sqrt{t - 2}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d t} [\frac{4 t}{3 {(t - 2)}^{\frac{1}{2}}}]$

Step 1.2

Since $\frac{4}{3}$ is constant with respect to $t$ , the derivative of $\frac{4 t}{3 {(t - 2)}^{\frac{1}{2}}}$ with respect to $t$ is $\frac{4}{3} \frac{d}{d t} [\frac{t}{{(t - 2)}^{\frac{1}{2}}}]$ .

$\frac{4}{3} \frac{d}{d t} [\frac{t}{{(t - 2)}^{\frac{1}{2}}}]$

Step 2

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t - 2)}^{\frac{1}{2}}]}{{({(t - 2)}^{\frac{1}{2}})}^{2}}$

Step 3

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

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Step 3.1

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t - 2)}^{\frac{1}{2}}]}{{(t - 2)}^{\frac{1}{2} \cdot 2}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t - 2)}^{\frac{1}{2}}]}{{(t - 2)}^{\frac{1}{2} \cdot 2}}$

Step 3.2.2

Rewrite the expression.

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t - 2)}^{\frac{1}{2}}]}{{(t - 2)}^{1}}$

Step 4

Simplify.

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} \frac{d}{d t} [t] - t \frac{d}{d t} [{(t - 2)}^{\frac{1}{2}}]}{t - 2}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} \cdot 1 - t \frac{d}{d t} [{(t - 2)}^{\frac{1}{2}}]}{t - 2}$

Step 5.2

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t \frac{d}{d t} [{(t - 2)}^{\frac{1}{2}}]}{t - 2}$

Step 6

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 $g (t) = t - 2$ .

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Step 6.1

To apply the Chain Rule, set $u$ as $t - 2$ .

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d t} [t - 2])}{t - 2}$

Step 6.2

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d t} [t - 2])}{t - 2}$

Step 6.3

Replace all occurrences of $u$ with $t - 2$ .

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} {(t - 2)}^{\frac{1}{2} - 1} \frac{d}{d t} [t - 2])}{t - 2}$

Step 7

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} {(t - 2)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [t - 2])}{t - 2}$

Step 8

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} {(t - 2)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [t - 2])}{t - 2}$

Step 9

Combine the numerators over the common denominator.

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} {(t - 2)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [t - 2])}{t - 2}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} {(t - 2)}^{\frac{1 - 2}{2}} \frac{d}{d t} [t - 2])}{t - 2}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} {(t - 2)}^{\frac{- 1}{2}} \frac{d}{d t} [t - 2])}{t - 2}$

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2} {(t - 2)}^{- \frac{1}{2}} \frac{d}{d t} [t - 2])}{t - 2}$

Step 11.2

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{{(t - 2)}^{- \frac{1}{2}}}{2} \frac{d}{d t} [t - 2])}{t - 2}$

Step 11.3

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - t (\frac{1}{2 {(t - 2)}^{\frac{1}{2}}} \frac{d}{d t} [t - 2])}{t - 2}$

Step 11.4

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - \frac{t}{2 {(t - 2)}^{\frac{1}{2}}} \frac{d}{d t} [t - 2]}{t - 2}$

Step 12

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - \frac{t}{2 {(t - 2)}^{\frac{1}{2}}} (\frac{d}{d t} [t] + \frac{d}{d t} [- 2])}{t - 2}$

Step 13

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

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - \frac{t}{2 {(t - 2)}^{\frac{1}{2}}} (1 + \frac{d}{d t} [- 2])}{t - 2}$

Step 14

Since $- 2$ is constant with respect to $t$ , the derivative of $- 2$ with respect to $t$ is $0$ .

$\frac{4}{3} \cdot \frac{{(t - 2)}^{\frac{1}{2}} - \frac{t}{2 {(t - 2)}^{\frac{1}{2}}} (1 + 0)}{t - 2}$

Step 15

Simplify the expression.

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Step 15.1

Add $1$ and $0$ .

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

Step 15.2

Multiply $- 1$ by $1$ .

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

Step 16

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

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

Step 17

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

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

Step 18

Combine the numerators over the common denominator.

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

Step 19

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

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Step 19.1

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

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

Step 19.2

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

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

Step 19.3

Combine the numerators over the common denominator.

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

Step 19.4

Add $1$ and $1$ .

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

Step 19.5

Divide $2$ by $2$ .

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

Step 20

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

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

Step 21

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

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

Step 22

Rewrite $\frac{\frac{2 (t - 2) - t}{2 {(t - 2)}^{\frac{1}{2}}}}{t - 2}$ as a product.

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

Step 23

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

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

Step 24

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

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

Step 25

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

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

Step 26

Simplify the expression.

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Step 26.1

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

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

Step 26.2

Combine the numerators over the common denominator.

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

Step 26.3

Add $2$ and $1$ .

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

Step 27

Multiply $\frac{4}{3}$ by $\frac{2 (t - 2) - t}{2 {(t - 2)}^{\frac{3}{2}}}$ .

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

Step 28

Multiply $2$ by $3$ .

$\frac{4 (2 (t - 2) - t)}{6 {(t - 2)}^{\frac{3}{2}}}$

Step 29

Factor $2$ out of $4 (2 (t - 2) - t)$ .

$\frac{2 (2 (2 (t - 2) - t))}{6 {(t - 2)}^{\frac{3}{2}}}$

Step 30

Cancel the common factors.

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Step 30.1

Factor $2$ out of $6 {(t - 2)}^{\frac{3}{2}}$ .

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

Step 30.2

Cancel the common factor.

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

Step 30.3

Rewrite the expression.

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

Step 31

Simplify.

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Step 31.1

Apply the distributive property.

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

Step 31.2

Apply the distributive property.

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

Step 31.3

Simplify the numerator.

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Step 31.3.1

Simplify each term.

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Step 31.3.1.1

Multiply $2$ by $2$ .

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

Step 31.3.1.2

Multiply $2 (2 \cdot - 2)$ .

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Step 31.3.1.2.1

Multiply $2$ by $- 2$ .

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

Step 31.3.1.2.2

Multiply $2$ by $- 4$ .

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

Step 31.3.1.3

Multiply $- 1$ by $2$ .

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

Step 31.3.2

Subtract $2 t$ from $4 t$ .

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

Step 31.4

Factor $2$ out of $2 t - 8$ .

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Step 31.4.1

Factor $2$ out of $2 t$ .

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

Step 31.4.2

Factor $2$ out of $- 8$ .

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

Step 31.4.3

Factor $2$ out of $2 t + 2 \cdot - 4$ .

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