Calculus Examples

$s = {(11 t - t^{2})}^{\frac{2}{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} ({(11 t - t^{2})}^{\frac{2}{3}})$

Step 2

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

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{2}{3}}$ and $g (t) = 11 t - t^{2}$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $11 t - t^{2}$ .

$\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d t} [11 t - t^{2}]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d t} [11 t - t^{2}]$

Step 3.1.3

Replace all occurrences of $u$ with $11 t - t^{2}$ .

$\frac{2}{3} {(11 t - t^{2})}^{\frac{2}{3} - 1} \frac{d}{d t} [11 t - t^{2}]$

Step 3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} {(11 t - t^{2})}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d t} [11 t - t^{2}]$

Step 3.3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{2}{3} {(11 t - t^{2})}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d t} [11 t - t^{2}]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{2}{3} {(11 t - t^{2})}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d t} [11 t - t^{2}]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} {(11 t - t^{2})}^{\frac{2 - 3}{3}} \frac{d}{d t} [11 t - t^{2}]$

Step 3.5.2

Subtract $3$ from $2$ .

$\frac{2}{3} {(11 t - t^{2})}^{\frac{- 1}{3}} \frac{d}{d t} [11 t - t^{2}]$

Step 3.6

Combine fractions.

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Step 3.6.1

Move the negative in front of the fraction.

$\frac{2}{3} {(11 t - t^{2})}^{- \frac{1}{3}} \frac{d}{d t} [11 t - t^{2}]$

Step 3.6.2

Combine $\frac{2}{3}$ and ${(11 t - t^{2})}^{- \frac{1}{3}}$ .

$\frac{2 {(11 t - t^{2})}^{- \frac{1}{3}}}{3} \frac{d}{d t} [11 t - t^{2}]$

Step 3.6.3

Move ${(11 t - t^{2})}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} \frac{d}{d t} [11 t - t^{2}]$

Step 3.7

By the Sum Rule, the derivative of $11 t - t^{2}$ with respect to $t$ is $\frac{d}{d t} [11 t] + \frac{d}{d t} [- t^{2}]$ .

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (\frac{d}{d t} [11 t] + \frac{d}{d t} [- t^{2}])$

Step 3.8

Since $11$ is constant with respect to $t$ , the derivative of $11 t$ with respect to $t$ is $11 \frac{d}{d t} [t]$ .

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (11 \frac{d}{d t} [t] + \frac{d}{d t} [- t^{2}])$

Step 3.9

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

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (11 \cdot 1 + \frac{d}{d t} [- t^{2}])$

Step 3.10

Multiply $11$ by $1$ .

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (11 + \frac{d}{d t} [- t^{2}])$

Step 3.11

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

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (11 - \frac{d}{d t} [t^{2}])$

Step 3.12

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

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (11 - (2 t))$

Step 3.13

Multiply $2$ by $- 1$ .

$\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (11 - 2 t)$

Step 3.14

Simplify.

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Step 3.14.1

Reorder the factors of $\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}} (11 - 2 t)$ .

$(11 - 2 t) \frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}}$

Step 3.14.2

Multiply $11 - 2 t$ by $\frac{2}{3 {(11 t - t^{2})}^{\frac{1}{3}}}$ .

$\frac{(11 - 2 t) \cdot 2}{3 {(11 t - t^{2})}^{\frac{1}{3}}}$

Step 3.14.3

Move $2$ to the left of $11 - 2 t$ .

$\frac{2 (11 - 2 t)}{3 {(11 t - t^{2})}^{\frac{1}{3}}}$

Step 4

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

$s' = \frac{2 (11 - 2 t)}{3 {(11 t - t^{2})}^{\frac{1}{3}}}$

Step 5

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

$\frac{d s}{d t} = \frac{2 (11 - 2 t)}{3 {(11 t - t^{2})}^{\frac{1}{3}}}$