Calculus Examples

$g (t) = \frac{t^{2}}{t - 1}$

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 $f (t) = t^{2}$ and $g (t) = t - 1$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Move $2$ to the left of $t - 1$ .

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

Step 2.3

By the Sum Rule, the derivative of $t - 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 1]$ .

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

Step 2.4

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

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

Step 2.5

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $1$ and $0$ .

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

Step 2.6.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Move $t$ .

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

Step 3.3.1.1.2

Multiply $t$ by $t$ .

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

Step 3.3.1.2

Multiply $2$ by $- 1$ .

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

Step 3.3.2

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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