Calculus Examples

$f (t) = \frac{t^{2} + 2}{2 t - 7}$

Step 1

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 $f (t) = t^{2} + 2$ and $g (t) = 2 t - 7$ .

$\frac{(2 t - 7) \frac{d}{d t} [t^{2} + 2] - (t^{2} + 2) \frac{d}{d t} [2 t - 7]}{{(2 t - 7)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(2 t - 7) (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [2]) - (t^{2} + 2) \frac{d}{d t} [2 t - 7]}{{(2 t - 7)}^{2}}$

Step 2.2

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

$\frac{(2 t - 7) (2 t + \frac{d}{d t} [2]) - (t^{2} + 2) \frac{d}{d t} [2 t - 7]}{{(2 t - 7)}^{2}}$

Step 2.3

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$\frac{(2 t - 7) (2 t + 0) - (t^{2} + 2) \frac{d}{d t} [2 t - 7]}{{(2 t - 7)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 t$ and $0$ .

$\frac{(2 t - 7) (2 t) - (t^{2} + 2) \frac{d}{d t} [2 t - 7]}{{(2 t - 7)}^{2}}$

Step 2.4.2

Move $2$ to the left of $2 t - 7$ .

$\frac{2 \cdot (2 t - 7) t - (t^{2} + 2) \frac{d}{d t} [2 t - 7]}{{(2 t - 7)}^{2}}$

Step 2.5

By the Sum Rule, the derivative of $2 t - 7$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [- 7]$ .

$\frac{2 (2 t - 7) t - (t^{2} + 2) (\frac{d}{d t} [2 t] + \frac{d}{d t} [- 7])}{{(2 t - 7)}^{2}}$

Step 2.6

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

$\frac{2 (2 t - 7) t - (t^{2} + 2) (2 \frac{d}{d t} [t] + \frac{d}{d t} [- 7])}{{(2 t - 7)}^{2}}$

Step 2.7

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

$\frac{2 (2 t - 7) t - (t^{2} + 2) (2 \cdot 1 + \frac{d}{d t} [- 7])}{{(2 t - 7)}^{2}}$

Step 2.8

Multiply $2$ by $1$ .

$\frac{2 (2 t - 7) t - (t^{2} + 2) (2 + \frac{d}{d t} [- 7])}{{(2 t - 7)}^{2}}$

Step 2.9

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7$ with respect to $t$ is $0$ .

$\frac{2 (2 t - 7) t - (t^{2} + 2) (2 + 0)}{{(2 t - 7)}^{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2$ and $0$ .

$\frac{2 (2 t - 7) t - (t^{2} + 2) \cdot 2}{{(2 t - 7)}^{2}}$

Step 2.10.2

Multiply $2$ by $- 1$ .

$\frac{2 (2 t - 7) t - 2 (t^{2} + 2)}{{(2 t - 7)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(2 (2 t) + 2 \cdot - 7) t - 2 (t^{2} + 2)}{{(2 t - 7)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{2 (2 t) t + 2 \cdot - 7 t - 2 (t^{2} + 2)}{{(2 t - 7)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{2 (2 t) t + 2 \cdot - 7 t - 2 t^{2} - 2 \cdot 2}{{(2 t - 7)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Move $t$ .

$\frac{2 (2 (t \cdot t)) + 2 \cdot - 7 t - 2 t^{2} - 2 \cdot 2}{{(2 t - 7)}^{2}}$

Step 3.4.1.1.2

Multiply $t$ by $t$ .

$\frac{2 (2 t^{2}) + 2 \cdot - 7 t - 2 t^{2} - 2 \cdot 2}{{(2 t - 7)}^{2}}$

Step 3.4.1.2

Multiply $2$ by $2$ .

$\frac{4 t^{2} + 2 \cdot - 7 t - 2 t^{2} - 2 \cdot 2}{{(2 t - 7)}^{2}}$

Step 3.4.1.3

Multiply $2$ by $- 7$ .

$\frac{4 t^{2} - 14 t - 2 t^{2} - 2 \cdot 2}{{(2 t - 7)}^{2}}$

Step 3.4.1.4

Multiply $- 2$ by $2$ .

$\frac{4 t^{2} - 14 t - 2 t^{2} - 4}{{(2 t - 7)}^{2}}$

Step 3.4.2

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

$\frac{2 t^{2} - 14 t - 4}{{(2 t - 7)}^{2}}$

Step 3.5

Factor $2$ out of $2 t^{2} - 14 t - 4$ .

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Step 3.5.1

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

$\frac{2 (t^{2}) - 14 t - 4}{{(2 t - 7)}^{2}}$

Step 3.5.2

Factor $2$ out of $- 14 t$ .

$\frac{2 (t^{2}) + 2 (- 7 t) - 4}{{(2 t - 7)}^{2}}$

Step 3.5.3

Factor $2$ out of $- 4$ .

$\frac{2 t^{2} + 2 (- 7 t) + 2 \cdot - 2}{{(2 t - 7)}^{2}}$

Step 3.5.4

Factor $2$ out of $2 t^{2} + 2 (- 7 t)$ .

$\frac{2 (t^{2} - 7 t) + 2 \cdot - 2}{{(2 t - 7)}^{2}}$

Step 3.5.5

Factor $2$ out of $2 (t^{2} - 7 t) + 2 \cdot - 2$ .

$\frac{2 (t^{2} - 7 t - 2)}{{(2 t - 7)}^{2}}$