Calculus Examples

$t \sqrt{10 - 2 t}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 - 2 t}$ as ${(10 - 2 t)}^{\frac{1}{2}}$ .

$\frac{d}{d t} [t {(10 - 2 t)}^{\frac{1}{2}}]$

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 $g (t) = {(10 - 2 t)}^{\frac{1}{2}}$ .

$t \frac{d}{d t} [{(10 - 2 t)}^{\frac{1}{2}}] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

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 $g (t) = 10 - 2 t$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $10 - 2 t$ .

$t (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 3.2

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

$t (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 3.3

Replace all occurrences of $u$ with $10 - 2 t$ .

$t (\frac{1}{2} {(10 - 2 t)}^{\frac{1}{2} - 1} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 4

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

$t (\frac{1}{2} {(10 - 2 t)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 5

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

$t (\frac{1}{2} {(10 - 2 t)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 6

Combine the numerators over the common denominator.

$t (\frac{1}{2} {(10 - 2 t)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$t (\frac{1}{2} {(10 - 2 t)}^{\frac{1 - 2}{2}} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 7.2

Subtract $2$ from $1$ .

$t (\frac{1}{2} {(10 - 2 t)}^{\frac{- 1}{2}} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$t (\frac{1}{2} {(10 - 2 t)}^{- \frac{1}{2}} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 8.2

Combine $\frac{1}{2}$ and ${(10 - 2 t)}^{- \frac{1}{2}}$ .

$t (\frac{{(10 - 2 t)}^{- \frac{1}{2}}}{2} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 8.3

Move ${(10 - 2 t)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$t (\frac{1}{2 {(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [10 - 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 8.4

Combine $\frac{1}{2 {(10 - 2 t)}^{\frac{1}{2}}}$ and $t$ .

$\frac{t}{2 {(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [10 - 2 t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 9

By the Sum Rule, the derivative of $10 - 2 t$ with respect to $t$ is $\frac{d}{d t} [10] + \frac{d}{d t} [- 2 t]$ .

$\frac{t}{2 {(10 - 2 t)}^{\frac{1}{2}}} (\frac{d}{d t} [10] + \frac{d}{d t} [- 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 10

Since $10$ is constant with respect to $t$ , the derivative of $10$ with respect to $t$ is $0$ .

$\frac{t}{2 {(10 - 2 t)}^{\frac{1}{2}}} (0 + \frac{d}{d t} [- 2 t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 11

Add $0$ and $\frac{d}{d t} [- 2 t]$ .

$\frac{t}{2 {(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [- 2 t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 12

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

$\frac{t}{2 {(10 - 2 t)}^{\frac{1}{2}}} (- 2 \frac{d}{d t} [t]) + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 13

Simplify terms.

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Step 13.1

Combine $- 2$ and $\frac{t}{2 {(10 - 2 t)}^{\frac{1}{2}}}$ .

$\frac{- 2 t}{2 {(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 13.2

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

$\frac{2 (- t)}{2 {(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 14

Cancel the common factors.

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Step 14.1

Factor $2$ out of $2 {(10 - 2 t)}^{\frac{1}{2}}$ .

$\frac{2 (- t)}{2 ({(10 - 2 t)}^{\frac{1}{2}})} \frac{d}{d t} [t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 14.2

Cancel the common factor.

$\frac{2 (- t)}{2 {(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 14.3

Rewrite the expression.

$\frac{- t}{{(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 15

Move the negative in front of the fraction.

$- \frac{t}{{(10 - 2 t)}^{\frac{1}{2}}} \frac{d}{d t} [t] + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 16

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

$- \frac{t}{{(10 - 2 t)}^{\frac{1}{2}}} \cdot 1 + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 17

Multiply $- 1$ by $1$ .

$- \frac{t}{{(10 - 2 t)}^{\frac{1}{2}}} + {(10 - 2 t)}^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 18

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

$- \frac{t}{{(10 - 2 t)}^{\frac{1}{2}}} + {(10 - 2 t)}^{\frac{1}{2}} \cdot 1$

Step 19

Multiply ${(10 - 2 t)}^{\frac{1}{2}}$ by $1$ .

$- \frac{t}{{(10 - 2 t)}^{\frac{1}{2}}} + {(10 - 2 t)}^{\frac{1}{2}}$

Step 20

To write ${(10 - 2 t)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(10 - 2 t)}^{\frac{1}{2}}}{{(10 - 2 t)}^{\frac{1}{2}}}$ .

$- \frac{t}{{(10 - 2 t)}^{\frac{1}{2}}} + \frac{{(10 - 2 t)}^{\frac{1}{2}} {(10 - 2 t)}^{\frac{1}{2}}}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 21

Combine the numerators over the common denominator.

$\frac{- t + {(10 - 2 t)}^{\frac{1}{2}} {(10 - 2 t)}^{\frac{1}{2}}}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 22

Multiply ${(10 - 2 t)}^{\frac{1}{2}}$ by ${(10 - 2 t)}^{\frac{1}{2}}$ by adding the exponents.

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Step 22.1

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

$\frac{- t + {(10 - 2 t)}^{\frac{1}{2} + \frac{1}{2}}}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 22.2

Combine the numerators over the common denominator.

$\frac{- t + {(10 - 2 t)}^{\frac{1 + 1}{2}}}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 22.3

Add $1$ and $1$ .

$\frac{- t + {(10 - 2 t)}^{\frac{2}{2}}}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 22.4

Divide $2$ by $2$ .

$\frac{- t + {(10 - 2 t)}^{1}}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 23

Simplify ${(10 - 2 t)}^{1}$ .

$\frac{- t + 10 - 2 t}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 24

Subtract $2 t$ from $- t$ .

$\frac{- 3 t + 10}{{(10 - 2 t)}^{\frac{1}{2}}}$

Step 25

Reorder terms.

$\frac{- 3 t + 10}{{(- 2 t + 10)}^{\frac{1}{2}}}$