Calculus Examples

$\frac{18 t}{t^{2} + 81}$

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 $f (t) = 18 t$ and $g (t) = t^{2} + 81$ .

$\frac{(t^{2} + 81) \frac{d}{d t} [18 t] - 1 \cdot 18 t \frac{d}{d t} [t^{2} + 81]}{{(t^{2} + 81)}^{2}}$

Step 2

Differentiate.

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Step 2.1

Since $18$ is constant with respect to $t$ , the derivative of $18 t$ with respect to $t$ is $18 \frac{d}{d t} [t]$ .

$\frac{(t^{2} + 81) (18 \frac{d}{d t} [t]) - 1 \cdot 18 t \frac{d}{d t} [t^{2} + 81]}{{(t^{2} + 81)}^{2}}$

Step 2.2

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

$\frac{(t^{2} + 81) (18 \cdot 1) - 1 \cdot 18 t \frac{d}{d t} [t^{2} + 81]}{{(t^{2} + 81)}^{2}}$

Step 2.3

Multiply $18$ by $1$ .

$\frac{(t^{2} + 81) \cdot 18 - 1 \cdot 18 t \frac{d}{d t} [t^{2} + 81]}{{(t^{2} + 81)}^{2}}$

Step 2.4

By the Sum Rule, the derivative of $t^{2} + 81$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [81]$ .

$\frac{(t^{2} + 81) \cdot 18 - 1 \cdot 18 t (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [81])}{{(t^{2} + 81)}^{2}}$

Step 2.5

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

$\frac{(t^{2} + 81) \cdot 18 - 1 \cdot 18 t (2 t + \frac{d}{d t} [81])}{{(t^{2} + 81)}^{2}}$

Step 2.6

Since $81$ is constant with respect to $t$ , the derivative of $81$ with respect to $t$ is $0$ .

$\frac{(t^{2} + 81) \cdot 18 - 1 \cdot 18 t (2 t + 0)}{{(t^{2} + 81)}^{2}}$

Step 2.7

Add $2 t$ and $0$ .

$\frac{(t^{2} + 81) \cdot 18 - 1 \cdot 18 t (2 t)}{{(t^{2} + 81)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{t^{2} \cdot 18 + 81 \cdot 18 - 1 \cdot 18 t (2 t)}{{(t^{2} + 81)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Move $18$ to the left of $t^{2}$ .

$\frac{18 \cdot t^{2} + 81 \cdot 18 - 1 \cdot 18 t (2 t)}{{(t^{2} + 81)}^{2}}$

Step 3.2.1.2

Multiply $81$ by $18$ .

$\frac{18 t^{2} + 1458 - 1 \cdot 18 t (2 t)}{{(t^{2} + 81)}^{2}}$

Step 3.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{18 t^{2} + 1458 - 1 \cdot 18 \cdot 2 t \cdot t}{{(t^{2} + 81)}^{2}}$

Step 3.2.1.4

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

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Step 3.2.1.4.1

Move $t$ .

$\frac{18 t^{2} + 1458 - 1 \cdot 18 \cdot 2 (t \cdot t)}{{(t^{2} + 81)}^{2}}$

Step 3.2.1.4.2

Multiply $t$ by $t$ .

$\frac{18 t^{2} + 1458 - 1 \cdot 18 \cdot 2 t^{2}}{{(t^{2} + 81)}^{2}}$

Step 3.2.1.5

Multiply $- 1 \cdot 18 \cdot 2$ .

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Step 3.2.1.5.1

Multiply $- 1$ by $18$ .

$\frac{18 t^{2} + 1458 - 18 \cdot 2 t^{2}}{{(t^{2} + 81)}^{2}}$

Step 3.2.1.5.2

Multiply $- 18$ by $2$ .

$\frac{18 t^{2} + 1458 - 36 t^{2}}{{(t^{2} + 81)}^{2}}$

Step 3.2.2

Subtract $36 t^{2}$ from $18 t^{2}$ .

$\frac{- 18 t^{2} + 1458}{{(t^{2} + 81)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Factor $18$ out of $- 18 t^{2} + 1458$ .

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Step 3.3.1.1

Factor $18$ out of $- 18 t^{2}$ .

$\frac{18 (- t^{2}) + 1458}{{(t^{2} + 81)}^{2}}$

Step 3.3.1.2

Factor $18$ out of $1458$ .

$\frac{18 (- t^{2}) + 18 (81)}{{(t^{2} + 81)}^{2}}$

Step 3.3.1.3

Factor $18$ out of $18 (- t^{2}) + 18 (81)$ .

$\frac{18 (- t^{2} + 81)}{{(t^{2} + 81)}^{2}}$

Step 3.3.2

Rewrite $81$ as $9^{2}$ .

$\frac{18 (- t^{2} + 9^{2})}{{(t^{2} + 81)}^{2}}$

Step 3.3.3

Reorder $- t^{2}$ and $9^{2}$ .

$\frac{18 (9^{2} - t^{2})}{{(t^{2} + 81)}^{2}}$

Step 3.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = t$ .

$\frac{18 (9 + t) (9 - t)}{{(t^{2} + 81)}^{2}}$