Calculus Examples

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

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

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

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 $n = 2$ .

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

Step 2.2

Move $2$ to the left of $t^{2} + t - 2$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Add $2 t + 1$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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 $t^{2}$ by $t$ by adding the exponents.

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Step 3.4.1.1.1

Move $t$ .

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

Step 3.4.1.1.2

Multiply $t$ by $t^{2}$ .

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Step 3.4.1.1.2.1

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

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

Step 3.4.1.1.2.2

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

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

Step 3.4.1.1.3

Add $1$ and $2$ .

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

Step 3.4.1.2

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

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Step 3.4.1.2.1

Move $t$ .

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

Step 3.4.1.2.2

Multiply $t$ by $t$ .

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

Step 3.4.1.3

Multiply $2$ by $- 2$ .

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

Step 3.4.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.4.1.5

Multiply $t^{2}$ by $t$ by adding the exponents.

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Step 3.4.1.5.1

Move $t$ .

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

Step 3.4.1.5.2

Multiply $t$ by $t^{2}$ .

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Step 3.4.1.5.2.1

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

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

Step 3.4.1.5.2.2

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

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

Step 3.4.1.5.3

Add $1$ and $2$ .

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

Step 3.4.1.6

Multiply $- 1$ by $2$ .

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

Step 3.4.1.7

Multiply $- 1$ by $1$ .

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

Step 3.4.2

Combine the opposite terms in $2 t^{3} + 2 t^{2} - 4 t - 2 t^{3} - t^{2}$ .

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Step 3.4.2.1

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

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

Step 3.4.2.2

Add $2 t^{2} - 4 t$ and $0$ .

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

Step 3.4.3

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Factor $t$ out of $- 4 t$ .

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

Step 3.5.3

Factor $t$ out of $t \cdot t + t \cdot - 4$ .

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

Step 3.6

Simplify the denominator.

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Step 3.6.1

Factor $t^{2} + t - 2$ using the AC method.

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Step 3.6.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.6.1.2

Write the factored form using these integers.

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

Step 3.6.2

Apply the product rule to $(t - 1) (t + 2)$ .

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