Calculus Examples

$\frac{500 t}{2 t^{2} + 25}$

Step 1

Since $500$ is constant with respect to $t$ , the derivative of $\frac{500 t}{2 t^{2} + 25}$ with respect to $t$ is $500 \frac{d}{d t} [\frac{t}{2 t^{2} + 25}]$ .

$500 \frac{d}{d t} [\frac{t}{2 t^{2} + 25}]$

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 $g (t) = 2 t^{2} + 25$ .

$500 \frac{(2 t^{2} + 25) \frac{d}{d t} [t] - t \frac{d}{d t} [2 t^{2} + 25]}{{(2 t^{2} + 25)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$500 \frac{(2 t^{2} + 25) \cdot 1 - t \frac{d}{d t} [2 t^{2} + 25]}{{(2 t^{2} + 25)}^{2}}$

Step 3.2

Multiply $2 t^{2} + 25$ by $1$ .

$500 \frac{2 t^{2} + 25 - t \frac{d}{d t} [2 t^{2} + 25]}{{(2 t^{2} + 25)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $2 t^{2} + 25$ with respect to $t$ is $\frac{d}{d t} [2 t^{2}] + \frac{d}{d t} [25]$ .

$500 \frac{2 t^{2} + 25 - t (\frac{d}{d t} [2 t^{2}] + \frac{d}{d t} [25])}{{(2 t^{2} + 25)}^{2}}$

Step 3.4

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

$500 \frac{2 t^{2} + 25 - t (2 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [25])}{{(2 t^{2} + 25)}^{2}}$

Step 3.5

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

$500 \frac{2 t^{2} + 25 - t (2 (2 t) + \frac{d}{d t} [25])}{{(2 t^{2} + 25)}^{2}}$

Step 3.6

Multiply $2$ by $2$ .

$500 \frac{2 t^{2} + 25 - t (4 t + \frac{d}{d t} [25])}{{(2 t^{2} + 25)}^{2}}$

Step 3.7

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

$500 \frac{2 t^{2} + 25 - t (4 t + 0)}{{(2 t^{2} + 25)}^{2}}$

Step 3.8

Simplify the expression.

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Step 3.8.1

Add $4 t$ and $0$ .

$500 \frac{2 t^{2} + 25 - t (4 t)}{{(2 t^{2} + 25)}^{2}}$

Step 3.8.2

Multiply $4$ by $- 1$ .

$500 \frac{2 t^{2} + 25 - 4 t \cdot t}{{(2 t^{2} + 25)}^{2}}$

Step 4

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

$500 \frac{2 t^{2} + 25 - 4 (t^{1} t)}{{(2 t^{2} + 25)}^{2}}$

Step 5

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

$500 \frac{2 t^{2} + 25 - 4 (t^{1} t^{1})}{{(2 t^{2} + 25)}^{2}}$

Step 6

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

$500 \frac{2 t^{2} + 25 - 4 t^{1 + 1}}{{(2 t^{2} + 25)}^{2}}$

Step 7

Add $1$ and $1$ .

$500 \frac{2 t^{2} + 25 - 4 t^{2}}{{(2 t^{2} + 25)}^{2}}$

Step 8

Subtract $4 t^{2}$ from $2 t^{2}$ .

$500 \frac{- 2 t^{2} + 25}{{(2 t^{2} + 25)}^{2}}$

Step 9

Combine $500$ and $\frac{- 2 t^{2} + 25}{{(2 t^{2} + 25)}^{2}}$ .

$\frac{500 (- 2 t^{2} + 25)}{{(2 t^{2} + 25)}^{2}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{500 (- 2 t^{2}) + 500 \cdot 25}{{(2 t^{2} + 25)}^{2}}$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $- 2$ by $500$ .

$\frac{- 1000 t^{2} + 500 \cdot 25}{{(2 t^{2} + 25)}^{2}}$

Step 10.2.2

Multiply $500$ by $25$ .

$\frac{- 1000 t^{2} + 12500}{{(2 t^{2} + 25)}^{2}}$

Step 10.3

Factor $500$ out of $- 1000 t^{2} + 12500$ .

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Step 10.3.1

Factor $500$ out of $- 1000 t^{2}$ .

$\frac{500 (- 2 t^{2}) + 12500}{{(2 t^{2} + 25)}^{2}}$

Step 10.3.2

Factor $500$ out of $12500$ .

$\frac{500 (- 2 t^{2}) + 500 (25)}{{(2 t^{2} + 25)}^{2}}$

Step 10.3.3

Factor $500$ out of $500 (- 2 t^{2}) + 500 (25)$ .

$\frac{500 (- 2 t^{2} + 25)}{{(2 t^{2} + 25)}^{2}}$

Step 10.4

Factor $- 1$ out of $- 2 t^{2}$ .

$\frac{500 (- (2 t^{2}) + 25)}{{(2 t^{2} + 25)}^{2}}$

Step 10.5

Rewrite $25$ as $- 1 (- 25)$ .

$\frac{500 (- (2 t^{2}) - 1 (- 25))}{{(2 t^{2} + 25)}^{2}}$

Step 10.6

Factor $- 1$ out of $- (2 t^{2}) - 1 (- 25)$ .

$\frac{500 (- (2 t^{2} - 25))}{{(2 t^{2} + 25)}^{2}}$

Step 10.7

Rewrite $- (2 t^{2} - 25)$ as $- 1 (2 t^{2} - 25)$ .

$\frac{500 (- 1 (2 t^{2} - 25))}{{(2 t^{2} + 25)}^{2}}$

Step 10.8

Move the negative in front of the fraction.

$- \frac{500 (2 t^{2} - 25)}{{(2 t^{2} + 25)}^{2}}$