Calculus Examples

$s = 10 \sqrt{t^{4} + 25} - 50$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t^{4} + 25}$ as ${(t^{4} + 25)}^{\frac{1}{2}}$ .

$s = 10 {(t^{4} + 25)}^{\frac{1}{2}} - 50$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (10 {(t^{4} + 25)}^{\frac{1}{2}} - 50)$

Step 3

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

$s'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $10 {(t^{4} + 25)}^{\frac{1}{2}} - 50$ with respect to $t$ is $\frac{d}{d t} [10 {(t^{4} + 25)}^{\frac{1}{2}}] + \frac{d}{d t} [- 50]$ .

$\frac{d}{d t} [10 {(t^{4} + 25)}^{\frac{1}{2}}] + \frac{d}{d t} [- 50]$

Step 4.2

Evaluate $\frac{d}{d t} [10 {(t^{4} + 25)}^{\frac{1}{2}}]$ .

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Step 4.2.1

Since $10$ is constant with respect to $t$ , the derivative of $10 {(t^{4} + 25)}^{\frac{1}{2}}$ with respect to $t$ is $10 \frac{d}{d t} [{(t^{4} + 25)}^{\frac{1}{2}}]$ .

$10 \frac{d}{d t} [{(t^{4} + 25)}^{\frac{1}{2}}] + \frac{d}{d t} [- 50]$

Step 4.2.2

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 $g (t) = t^{4} + 25$ .

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Step 4.2.2.1

To apply the Chain Rule, set $u$ as $t^{4} + 25$ .

$10 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d t} [t^{4} + 25]) + \frac{d}{d t} [- 50]$

Step 4.2.2.2

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

$10 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d t} [t^{4} + 25]) + \frac{d}{d t} [- 50]$

Step 4.2.2.3

Replace all occurrences of $u$ with $t^{4} + 25$ .

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1}{2} - 1} \frac{d}{d t} [t^{4} + 25]) + \frac{d}{d t} [- 50]$

Step 4.2.3

By the Sum Rule, the derivative of $t^{4} + 25$ with respect to $t$ is $\frac{d}{d t} [t^{4}] + \frac{d}{d t} [25]$ .

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1}{2} - 1} (\frac{d}{d t} [t^{4}] + \frac{d}{d t} [25])) + \frac{d}{d t} [- 50]$

Step 4.2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1}{2} - 1} (4 t^{3} + \frac{d}{d t} [25])) + \frac{d}{d t} [- 50]$

Step 4.2.5

Since $25$ is constant with respect to $t$ , the derivative of $25$ with respect to $t$ is $0$ .

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1}{2} - 1} (4 t^{3} + 0)) + \frac{d}{d t} [- 50]$

Step 4.2.6

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

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (4 t^{3} + 0)) + \frac{d}{d t} [- 50]$

Step 4.2.7

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

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (4 t^{3} + 0)) + \frac{d}{d t} [- 50]$

Step 4.2.8

Combine the numerators over the common denominator.

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1 - 1 \cdot 2}{2}} (4 t^{3} + 0)) + \frac{d}{d t} [- 50]$

Step 4.2.9

Simplify the numerator.

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Step 4.2.9.1

Multiply $- 1$ by $2$ .

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{1 - 2}{2}} (4 t^{3} + 0)) + \frac{d}{d t} [- 50]$

Step 4.2.9.2

Subtract $2$ from $1$ .

$10 (\frac{1}{2} {(t^{4} + 25)}^{\frac{- 1}{2}} (4 t^{3} + 0)) + \frac{d}{d t} [- 50]$

Step 4.2.10

Move the negative in front of the fraction.

$10 (\frac{1}{2} {(t^{4} + 25)}^{- \frac{1}{2}} (4 t^{3} + 0)) + \frac{d}{d t} [- 50]$

Step 4.2.11

Add $4 t^{3}$ and $0$ .

$10 (\frac{1}{2} {(t^{4} + 25)}^{- \frac{1}{2}} (4 t^{3})) + \frac{d}{d t} [- 50]$

Step 4.2.12

Combine $\frac{1}{2}$ and ${(t^{4} + 25)}^{- \frac{1}{2}}$ .

$10 (\frac{{(t^{4} + 25)}^{- \frac{1}{2}}}{2} (4 t^{3})) + \frac{d}{d t} [- 50]$

Step 4.2.13

Combine $4$ and $\frac{{(t^{4} + 25)}^{- \frac{1}{2}}}{2}$ .

$10 (\frac{4 {(t^{4} + 25)}^{- \frac{1}{2}}}{2} t^{3}) + \frac{d}{d t} [- 50]$

Step 4.2.14

Combine $\frac{4 {(t^{4} + 25)}^{- \frac{1}{2}}}{2}$ and $t^{3}$ .

$10 \frac{4 {(t^{4} + 25)}^{- \frac{1}{2}} t^{3}}{2} + \frac{d}{d t} [- 50]$

Step 4.2.15

Move ${(t^{4} + 25)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 \frac{4 t^{3}}{2 {(t^{4} + 25)}^{\frac{1}{2}}} + \frac{d}{d t} [- 50]$

Step 4.2.16

Factor $2$ out of $4 t^{3}$ .

$10 \frac{2 (2 t^{3})}{2 {(t^{4} + 25)}^{\frac{1}{2}}} + \frac{d}{d t} [- 50]$

Step 4.2.17

Cancel the common factors.

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Step 4.2.17.1

Factor $2$ out of $2 {(t^{4} + 25)}^{\frac{1}{2}}$ .

$10 \frac{2 (2 t^{3})}{2 ({(t^{4} + 25)}^{\frac{1}{2}})} + \frac{d}{d t} [- 50]$

Step 4.2.17.2

Cancel the common factor.

$10 \frac{2 (2 t^{3})}{2 {(t^{4} + 25)}^{\frac{1}{2}}} + \frac{d}{d t} [- 50]$

Step 4.2.17.3

Rewrite the expression.

$10 \frac{2 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}} + \frac{d}{d t} [- 50]$

Step 4.2.18

Combine $10$ and $\frac{2 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}}$ .

$\frac{10 (2 t^{3})}{{(t^{4} + 25)}^{\frac{1}{2}}} + \frac{d}{d t} [- 50]$

Step 4.2.19

Multiply $2$ by $10$ .

$\frac{20 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}} + \frac{d}{d t} [- 50]$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $- 50$ is constant with respect to $t$ , the derivative of $- 50$ with respect to $t$ is $0$ .

$\frac{20 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}} + 0$

Step 4.3.2

Add $\frac{20 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}}$ and $0$ .

$\frac{20 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}}$

Step 5

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

$s' = \frac{20 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}}$

Step 6

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

$\frac{d s}{d t} = \frac{20 t^{3}}{{(t^{4} + 25)}^{\frac{1}{2}}}$