Calculus Examples

$\frac{6 t^{2} - 6}{2 t - 4}$

Step 1

Cancel the common factor of $6 t^{2} - 6$ and $2 t - 4$ .

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Step 1.1

Factor $2$ out of $6 t^{2}$ .

$\frac{d}{d t} [\frac{2 (3 t^{2}) - 6}{2 t - 4}]$

Step 1.2

Factor $2$ out of $- 6$ .

$\frac{d}{d t} [\frac{2 (3 t^{2}) + 2 \cdot - 3}{2 t - 4}]$

Step 1.3

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

$\frac{d}{d t} [\frac{2 (3 t^{2} - 3)}{2 t - 4}]$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $2 t$ .

$\frac{d}{d t} [\frac{2 (3 t^{2} - 3)}{2 (t) - 4}]$

Step 1.4.2

Factor $2$ out of $- 4$ .

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

Step 1.4.3

Factor $2$ out of $2 t + 2 \cdot - 2$ .

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

Step 1.4.4

Cancel the common factor.

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

Step 1.4.5

Rewrite the expression.

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

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $3$ .

$\frac{(t - 2) (6 t + \frac{d}{d t} [- 3]) - (3 t^{2} - 3) \frac{d}{d t} [t - 2]}{{(t - 2)}^{2}}$

Step 3.5

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

$\frac{(t - 2) (6 t + 0) - (3 t^{2} - 3) \frac{d}{d t} [t - 2]}{{(t - 2)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $6 t$ and $0$ .

$\frac{(t - 2) (6 t) - (3 t^{2} - 3) \frac{d}{d t} [t - 2]}{{(t - 2)}^{2}}$

Step 3.6.2

Move $6$ to the left of $t - 2$ .

$\frac{6 \cdot (t - 2) t - (3 t^{2} - 3) \frac{d}{d t} [t - 2]}{{(t - 2)}^{2}}$

Step 3.7

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

$\frac{6 (t - 2) t - (3 t^{2} - 3) (\frac{d}{d t} [t] + \frac{d}{d t} [- 2])}{{(t - 2)}^{2}}$

Step 3.8

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

$\frac{6 (t - 2) t - (3 t^{2} - 3) (1 + \frac{d}{d t} [- 2])}{{(t - 2)}^{2}}$

Step 3.9

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

$\frac{6 (t - 2) t - (3 t^{2} - 3) (1 + 0)}{{(t - 2)}^{2}}$

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $1$ and $0$ .

$\frac{6 (t - 2) t - (3 t^{2} - 3) \cdot 1}{{(t - 2)}^{2}}$

Step 3.10.2

Multiply $- 1$ by $1$ .

$\frac{6 (t - 2) t - (3 t^{2} - 3)}{{(t - 2)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{(6 t + 6 \cdot - 2) t - (3 t^{2} - 3)}{{(t - 2)}^{2}}$

Step 4.2

Apply the distributive property.

$\frac{6 t \cdot t + 6 \cdot - 2 t - (3 t^{2} - 3)}{{(t - 2)}^{2}}$

Step 4.3

Apply the distributive property.

$\frac{6 t \cdot t + 6 \cdot - 2 t - (3 t^{2}) - - 3}{{(t - 2)}^{2}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

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

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Step 4.4.1.1.1

Move $t$ .

$\frac{6 (t \cdot t) + 6 \cdot - 2 t - (3 t^{2}) - - 3}{{(t - 2)}^{2}}$

Step 4.4.1.1.2

Multiply $t$ by $t$ .

$\frac{6 t^{2} + 6 \cdot - 2 t - (3 t^{2}) - - 3}{{(t - 2)}^{2}}$

Step 4.4.1.2

Multiply $6$ by $- 2$ .

$\frac{6 t^{2} - 12 t - (3 t^{2}) - - 3}{{(t - 2)}^{2}}$

Step 4.4.1.3

Multiply $3$ by $- 1$ .

$\frac{6 t^{2} - 12 t - 3 t^{2} - - 3}{{(t - 2)}^{2}}$

Step 4.4.1.4

Multiply $- 1$ by $- 3$ .

$\frac{6 t^{2} - 12 t - 3 t^{2} + 3}{{(t - 2)}^{2}}$

Step 4.4.2

Subtract $3 t^{2}$ from $6 t^{2}$ .

$\frac{3 t^{2} - 12 t + 3}{{(t - 2)}^{2}}$

Step 4.5

Factor $3$ out of $3 t^{2} - 12 t + 3$ .

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Step 4.5.1

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

$\frac{3 (t^{2}) - 12 t + 3}{{(t - 2)}^{2}}$

Step 4.5.2

Factor $3$ out of $- 12 t$ .

$\frac{3 (t^{2}) + 3 (- 4 t) + 3}{{(t - 2)}^{2}}$

Step 4.5.3

Factor $3$ out of $3$ .

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

Step 4.5.4

Factor $3$ out of $3 (t^{2}) + 3 (- 4 t)$ .

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

Step 4.5.5

Factor $3$ out of $3 (t^{2} - 4 t) + 3 (1)$ .

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