Calculus Examples

$q (t) = \frac{5 t}{t^{2} - 5 t - 3}$

Step 1

Since $5$ is constant with respect to $t$ , the derivative of $\frac{5 t}{t^{2} - 5 t - 3}$ with respect to $t$ is $5 \frac{d}{d t} [\frac{t}{t^{2} - 5 t - 3}]$ .

$5 \frac{d}{d t} [\frac{t}{t^{2} - 5 t - 3}]$

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 $g (t) = t^{2} - 5 t - 3$ .

$5 \frac{(t^{2} - 5 t - 3) \frac{d}{d t} [t] - t \frac{d}{d t} [t^{2} - 5 t - 3]}{{(t^{2} - 5 t - 3)}^{2}}$

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 $n = 1$ .

$5 \frac{(t^{2} - 5 t - 3) \cdot 1 - t \frac{d}{d t} [t^{2} - 5 t - 3]}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.2

Multiply $t^{2} - 5 t - 3$ by $1$ .

$5 \frac{t^{2} - 5 t - 3 - t \frac{d}{d t} [t^{2} - 5 t - 3]}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $t^{2} - 5 t - 3$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 5 t] + \frac{d}{d t} [- 3]$ .

$5 \frac{t^{2} - 5 t - 3 - t (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 5 t] + \frac{d}{d t} [- 3])}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.4

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

$5 \frac{t^{2} - 5 t - 3 - t (2 t + \frac{d}{d t} [- 5 t] + \frac{d}{d t} [- 3])}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.5

Since $- 5$ is constant with respect to $t$ , the derivative of $- 5 t$ with respect to $t$ is $- 5 \frac{d}{d t} [t]$ .

$5 \frac{t^{2} - 5 t - 3 - t (2 t - 5 \frac{d}{d t} [t] + \frac{d}{d t} [- 3])}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.6

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

$5 \frac{t^{2} - 5 t - 3 - t (2 t - 5 \cdot 1 + \frac{d}{d t} [- 3])}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.7

Multiply $- 5$ by $1$ .

$5 \frac{t^{2} - 5 t - 3 - t (2 t - 5 + \frac{d}{d t} [- 3])}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.8

Since $- 3$ is constant with respect to $t$ , the derivative of $- 3$ with respect to $t$ is $0$ .

$5 \frac{t^{2} - 5 t - 3 - t (2 t - 5 + 0)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.9

Combine fractions.

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Step 3.9.1

Add $2 t - 5$ and $0$ .

$5 \frac{t^{2} - 5 t - 3 - t (2 t - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 3.9.2

Combine $5$ and $\frac{t^{2} - 5 t - 3 - t (2 t - 5)}{{(t^{2} - 5 t - 3)}^{2}}$ .

$\frac{5 (t^{2} - 5 t - 3 - t (2 t - 5))}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{5 (t^{2} - 5 t - 3 - t (2 t) - t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.2

Apply the distributive property.

$\frac{5 t^{2} + 5 (- 5 t) + 5 \cdot - 3 + 5 (- t (2 t)) + 5 (- t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $- 5$ by $5$ .

$\frac{5 t^{2} - 25 t + 5 \cdot - 3 + 5 (- t (2 t)) + 5 (- t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.1.2

Multiply $5$ by $- 3$ .

$\frac{5 t^{2} - 25 t - 15 + 5 (- t (2 t)) + 5 (- t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.1.3

Multiply $t$ by $t$ by adding the exponents.

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Step 4.3.1.3.1

Move $t$ .

$\frac{5 t^{2} - 25 t - 15 + 5 (- (t \cdot t) \cdot 2) + 5 (- t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.1.3.2

Multiply $t$ by $t$ .

$\frac{5 t^{2} - 25 t - 15 + 5 (- t^{2} \cdot 2) + 5 (- t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.1.4

Multiply $2$ by $- 1$ .

$\frac{5 t^{2} - 25 t - 15 + 5 (- 2 t^{2}) + 5 (- t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.1.5

Multiply $- 2$ by $5$ .

$\frac{5 t^{2} - 25 t - 15 - 10 t^{2} + 5 (- t \cdot - 5)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.1.6

Multiply $- 5$ by $- 1$ .

$\frac{5 t^{2} - 25 t - 15 - 10 t^{2} + 5 (5 t)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.1.7

Multiply $5$ by $5$ .

$\frac{5 t^{2} - 25 t - 15 - 10 t^{2} + 25 t}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.2

Combine the opposite terms in $5 t^{2} - 25 t - 15 - 10 t^{2} + 25 t$ .

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Step 4.3.2.1

Add $- 25 t$ and $25 t$ .

$\frac{5 t^{2} - 15 - 10 t^{2} + 0}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.2.2

Add $5 t^{2} - 15 - 10 t^{2}$ and $0$ .

$\frac{5 t^{2} - 15 - 10 t^{2}}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.3.3

Subtract $10 t^{2}$ from $5 t^{2}$ .

$\frac{- 5 t^{2} - 15}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.4

Factor $5$ out of $- 5 t^{2} - 15$ .

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Step 4.4.1

Factor $5$ out of $- 5 t^{2}$ .

$\frac{5 (- t^{2}) - 15}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.4.2

Factor $5$ out of $- 15$ .

$\frac{5 (- t^{2}) + 5 (- 3)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.4.3

Factor $5$ out of $5 (- t^{2}) + 5 (- 3)$ .

$\frac{5 (- t^{2} - 3)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.5

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

$\frac{5 (- (t^{2}) - 3)}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.6

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{5 (- (t^{2}) - 1 (3))}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.7

Factor $- 1$ out of $- (t^{2}) - 1 (3)$ .

$\frac{5 (- (t^{2} + 3))}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.8

Rewrite $- (t^{2} + 3)$ as $- 1 (t^{2} + 3)$ .

$\frac{5 (- 1 (t^{2} + 3))}{{(t^{2} - 5 t - 3)}^{2}}$

Step 4.9

Move the negative in front of the fraction.

$- \frac{5 (t^{2} + 3)}{{(t^{2} - 5 t - 3)}^{2}}$