Calculus Examples

$\frac{70 t}{{(3 t^{2} + 8)}^{2}}$

Step 1

Since $70$ is constant with respect to $t$ , the derivative of $\frac{70 t}{{(3 t^{2} + 8)}^{2}}$ with respect to $t$ is $70 \frac{d}{d t} [\frac{t}{{(3 t^{2} + 8)}^{2}}]$ .

$70 \frac{d}{d t} [\frac{t}{{(3 t^{2} + 8)}^{2}}]$

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 $f (t) = t$ and $g (t) = {(3 t^{2} + 8)}^{2}$ .

$70 \frac{{(3 t^{2} + 8)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(3 t^{2} + 8)}^{2}]}{{({(3 t^{2} + 8)}^{2})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(3 t^{2} + 8)}^{2})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$70 \frac{{(3 t^{2} + 8)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(3 t^{2} + 8)}^{2}]}{{(3 t^{2} + 8)}^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$70 \frac{{(3 t^{2} + 8)}^{2} \frac{d}{d t} [t] - t \frac{d}{d t} [{(3 t^{2} + 8)}^{2}]}{{(3 t^{2} + 8)}^{4}}$

Step 3.2

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

$70 \frac{{(3 t^{2} + 8)}^{2} \cdot 1 - t \frac{d}{d t} [{(3 t^{2} + 8)}^{2}]}{{(3 t^{2} + 8)}^{4}}$

Step 3.3

Multiply ${(3 t^{2} + 8)}^{2}$ by $1$ .

$70 \frac{{(3 t^{2} + 8)}^{2} - t \frac{d}{d t} [{(3 t^{2} + 8)}^{2}]}{{(3 t^{2} + 8)}^{4}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = 3 t^{2} + 8$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $3 t^{2} + 8$ .

$70 \frac{{(3 t^{2} + 8)}^{2} - t (\frac{d}{d u} [u^{2}] \frac{d}{d t} [3 t^{2} + 8])}{{(3 t^{2} + 8)}^{4}}$

Step 4.2

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

$70 \frac{{(3 t^{2} + 8)}^{2} - t (2 u \frac{d}{d t} [3 t^{2} + 8])}{{(3 t^{2} + 8)}^{4}}$

Step 4.3

Replace all occurrences of $u$ with $3 t^{2} + 8$ .

$70 \frac{{(3 t^{2} + 8)}^{2} - t (2 (3 t^{2} + 8) \frac{d}{d t} [3 t^{2} + 8])}{{(3 t^{2} + 8)}^{4}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $2$ by $- 1$ .

$70 \frac{{(3 t^{2} + 8)}^{2} - 2 t ((3 t^{2} + 8) \frac{d}{d t} [3 t^{2} + 8])}{{(3 t^{2} + 8)}^{4}}$

Step 5.2

Factor $3 t^{2} + 8$ out of ${(3 t^{2} + 8)}^{2} - 2 t ((3 t^{2} + 8) \frac{d}{d t} [3 t^{2} + 8])$ .

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Step 5.2.1

Factor $3 t^{2} + 8$ out of ${(3 t^{2} + 8)}^{2}$ .

$70 \frac{(3 t^{2} + 8) (3 t^{2} + 8) - 2 t ((3 t^{2} + 8) \frac{d}{d t} [3 t^{2} + 8])}{{(3 t^{2} + 8)}^{4}}$

Step 5.2.2

Factor $3 t^{2} + 8$ out of $- 2 t ((3 t^{2} + 8) \frac{d}{d t} [3 t^{2} + 8])$ .

$70 \frac{(3 t^{2} + 8) (3 t^{2} + 8) + (3 t^{2} + 8) (- 2 t (\frac{d}{d t} [3 t^{2} + 8]))}{{(3 t^{2} + 8)}^{4}}$

Step 5.2.3

Factor $3 t^{2} + 8$ out of $(3 t^{2} + 8) (3 t^{2} + 8) + (3 t^{2} + 8) (- 2 t (\frac{d}{d t} [3 t^{2} + 8]))$ .

$70 \frac{(3 t^{2} + 8) (3 t^{2} + 8 - 2 t (\frac{d}{d t} [3 t^{2} + 8]))}{{(3 t^{2} + 8)}^{4}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $3 t^{2} + 8$ out of ${(3 t^{2} + 8)}^{4}$ .

$70 \frac{(3 t^{2} + 8) (3 t^{2} + 8 - 2 t (\frac{d}{d t} [3 t^{2} + 8]))}{(3 t^{2} + 8) {(3 t^{2} + 8)}^{3}}$

Step 6.2

Cancel the common factor.

$70 \frac{(3 t^{2} + 8) (3 t^{2} + 8 - 2 t (\frac{d}{d t} [3 t^{2} + 8]))}{(3 t^{2} + 8) {(3 t^{2} + 8)}^{3}}$

Step 6.3

Rewrite the expression.

$70 \frac{3 t^{2} + 8 - 2 t (\frac{d}{d t} [3 t^{2} + 8])}{{(3 t^{2} + 8)}^{3}}$

Step 7

By the Sum Rule, the derivative of $3 t^{2} + 8$ with respect to $t$ is $\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [8]$ .

$70 \frac{3 t^{2} + 8 - 2 t (\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [8])}{{(3 t^{2} + 8)}^{3}}$

Step 8

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

$70 \frac{3 t^{2} + 8 - 2 t (3 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [8])}{{(3 t^{2} + 8)}^{3}}$

Step 9

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

$70 \frac{3 t^{2} + 8 - 2 t (3 (2 t) + \frac{d}{d t} [8])}{{(3 t^{2} + 8)}^{3}}$

Step 10

Multiply $2$ by $3$ .

$70 \frac{3 t^{2} + 8 - 2 t (6 t + \frac{d}{d t} [8])}{{(3 t^{2} + 8)}^{3}}$

Step 11

Since $8$ is constant with respect to $t$ , the derivative of $8$ with respect to $t$ is $0$ .

$70 \frac{3 t^{2} + 8 - 2 t (6 t + 0)}{{(3 t^{2} + 8)}^{3}}$

Step 12

Simplify the expression.

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Step 12.1

Add $6 t$ and $0$ .

$70 \frac{3 t^{2} + 8 - 2 t (6 t)}{{(3 t^{2} + 8)}^{3}}$

Step 12.2

Multiply $6$ by $- 2$ .

$70 \frac{3 t^{2} + 8 - 12 t \cdot t}{{(3 t^{2} + 8)}^{3}}$

Step 13

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

$70 \frac{3 t^{2} + 8 - 12 (t^{1} t)}{{(3 t^{2} + 8)}^{3}}$

Step 14

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

$70 \frac{3 t^{2} + 8 - 12 (t^{1} t^{1})}{{(3 t^{2} + 8)}^{3}}$

Step 15

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

$70 \frac{3 t^{2} + 8 - 12 t^{1 + 1}}{{(3 t^{2} + 8)}^{3}}$

Step 16

Add $1$ and $1$ .

$70 \frac{3 t^{2} + 8 - 12 t^{2}}{{(3 t^{2} + 8)}^{3}}$

Step 17

Subtract $12 t^{2}$ from $3 t^{2}$ .

$70 \frac{- 9 t^{2} + 8}{{(3 t^{2} + 8)}^{3}}$

Step 18

Combine $70$ and $\frac{- 9 t^{2} + 8}{{(3 t^{2} + 8)}^{3}}$ .

$\frac{70 (- 9 t^{2} + 8)}{{(3 t^{2} + 8)}^{3}}$

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

$\frac{70 (- 9 t^{2}) + 70 \cdot 8}{{(3 t^{2} + 8)}^{3}}$

Step 19.2

Simplify each term.

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Step 19.2.1

Multiply $- 9$ by $70$ .

$\frac{- 630 t^{2} + 70 \cdot 8}{{(3 t^{2} + 8)}^{3}}$

Step 19.2.2

Multiply $70$ by $8$ .

$\frac{- 630 t^{2} + 560}{{(3 t^{2} + 8)}^{3}}$

Step 19.3

Factor $70$ out of $- 630 t^{2} + 560$ .

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Step 19.3.1

Factor $70$ out of $- 630 t^{2}$ .

$\frac{70 (- 9 t^{2}) + 560}{{(3 t^{2} + 8)}^{3}}$

Step 19.3.2

Factor $70$ out of $560$ .

$\frac{70 (- 9 t^{2}) + 70 (8)}{{(3 t^{2} + 8)}^{3}}$

Step 19.3.3

Factor $70$ out of $70 (- 9 t^{2}) + 70 (8)$ .

$\frac{70 (- 9 t^{2} + 8)}{{(3 t^{2} + 8)}^{3}}$

Step 19.4

Factor $- 1$ out of $- 9 t^{2}$ .

$\frac{70 (- (9 t^{2}) + 8)}{{(3 t^{2} + 8)}^{3}}$

Step 19.5

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{70 (- (9 t^{2}) - 1 (- 8))}{{(3 t^{2} + 8)}^{3}}$

Step 19.6

Factor $- 1$ out of $- (9 t^{2}) - 1 (- 8)$ .

$\frac{70 (- (9 t^{2} - 8))}{{(3 t^{2} + 8)}^{3}}$

Step 19.7

Rewrite $- (9 t^{2} - 8)$ as $- 1 (9 t^{2} - 8)$ .

$\frac{70 (- 1 (9 t^{2} - 8))}{{(3 t^{2} + 8)}^{3}}$

Step 19.8

Move the negative in front of the fraction.

$- \frac{70 (9 t^{2} - 8)}{{(3 t^{2} + 8)}^{3}}$