Calculus Examples

$r (t) = \frac{{(5 t - 6)}^{4}}{3 t^{2} + 4}$

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 $f (t) = {(5 t - 6)}^{4}$ and $g (t) = 3 t^{2} + 4$ .

$\frac{(3 t^{2} + 4) \frac{d}{d t} [{(5 t - 6)}^{4}] - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{4}$ and $g (t) = 5 t - 6$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $5 t - 6$ .

$\frac{(3 t^{2} + 4) (\frac{d}{d u} [u^{4}] \frac{d}{d t} [5 t - 6]) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$\frac{(3 t^{2} + 4) (4 u^{3} \frac{d}{d t} [5 t - 6]) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 2.3

Replace all occurrences of $u$ with $5 t - 6$ .

$\frac{(3 t^{2} + 4) (4 {(5 t - 6)}^{3} \frac{d}{d t} [5 t - 6]) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3

Differentiate.

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Step 3.1

Move $4$ to the left of $3 t^{2} + 4$ .

$\frac{4 \cdot (3 t^{2} + 4) {(5 t - 6)}^{3} \frac{d}{d t} [5 t - 6] - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.2

By the Sum Rule, the derivative of $5 t - 6$ with respect to $t$ is $\frac{d}{d t} [5 t] + \frac{d}{d t} [- 6]$ .

$\frac{4 (3 t^{2} + 4) {(5 t - 6)}^{3} (\frac{d}{d t} [5 t] + \frac{d}{d t} [- 6]) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.3

Since $5$ is constant with respect to $t$ , the derivative of $5 t$ with respect to $t$ is $5 \frac{d}{d t} [t]$ .

$\frac{4 (3 t^{2} + 4) {(5 t - 6)}^{3} (5 \frac{d}{d t} [t] + \frac{d}{d t} [- 6]) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.4

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

$\frac{4 (3 t^{2} + 4) {(5 t - 6)}^{3} (5 \cdot 1 + \frac{d}{d t} [- 6]) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.5

Multiply $5$ by $1$ .

$\frac{4 (3 t^{2} + 4) {(5 t - 6)}^{3} (5 + \frac{d}{d t} [- 6]) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.6

Since $- 6$ is constant with respect to $t$ , the derivative of $- 6$ with respect to $t$ is $0$ .

$\frac{4 (3 t^{2} + 4) {(5 t - 6)}^{3} (5 + 0) - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $5$ and $0$ .

$\frac{4 (3 t^{2} + 4) {(5 t - 6)}^{3} \cdot 5 - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.7.2

Multiply $5$ by $4$ .

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - {(5 t - 6)}^{4} \frac{d}{d t} [3 t^{2} + 4]}{{(3 t^{2} + 4)}^{2}}$

Step 3.8

By the Sum Rule, the derivative of $3 t^{2} + 4$ with respect to $t$ is $\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [4]$ .

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - {(5 t - 6)}^{4} (\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [4])}{{(3 t^{2} + 4)}^{2}}$

Step 3.9

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

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - {(5 t - 6)}^{4} (3 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [4])}{{(3 t^{2} + 4)}^{2}}$

Step 3.10

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

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - {(5 t - 6)}^{4} (3 (2 t) + \frac{d}{d t} [4])}{{(3 t^{2} + 4)}^{2}}$

Step 3.11

Multiply $2$ by $3$ .

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - {(5 t - 6)}^{4} (6 t + \frac{d}{d t} [4])}{{(3 t^{2} + 4)}^{2}}$

Step 3.12

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - {(5 t - 6)}^{4} (6 t + 0)}{{(3 t^{2} + 4)}^{2}}$

Step 3.13

Simplify the expression.

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Step 3.13.1

Add $6 t$ and $0$ .

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - {(5 t - 6)}^{4} (6 t)}{{(3 t^{2} + 4)}^{2}}$

Step 3.13.2

Multiply $6$ by $- 1$ .

$\frac{20 (3 t^{2} + 4) {(5 t - 6)}^{3} - 6 {(5 t - 6)}^{4} t}{{(3 t^{2} + 4)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{(20 (3 t^{2}) + 20 \cdot 4) {(5 t - 6)}^{3} - 6 {(5 t - 6)}^{4} t}{{(3 t^{2} + 4)}^{2}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Factor ${(5 t - 6)}^{3}$ out of $(20 \cdot 3 t^{2} + 20 \cdot 4) {(5 t - 6)}^{3} - 6 {(5 t - 6)}^{4} t$ .

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Step 4.2.1.1

Factor ${(5 t - 6)}^{3}$ out of $(20 \cdot 3 t^{2} + 20 \cdot 4) {(5 t - 6)}^{3}$ .

$\frac{{(5 t - 6)}^{3} (20 \cdot 3 t^{2} + 20 \cdot 4) - 6 {(5 t - 6)}^{4} t}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.1.2

Factor ${(5 t - 6)}^{3}$ out of $- 6 {(5 t - 6)}^{4} t$ .

$\frac{{(5 t - 6)}^{3} (20 \cdot 3 t^{2} + 20 \cdot 4) + {(5 t - 6)}^{3} (- 6 (5 t - 6) t)}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.1.3

Factor ${(5 t - 6)}^{3}$ out of ${(5 t - 6)}^{3} (20 \cdot 3 t^{2} + 20 \cdot 4) + {(5 t - 6)}^{3} (- 6 (5 t - 6) t)$ .

$\frac{{(5 t - 6)}^{3} (20 \cdot 3 t^{2} + 20 \cdot 4 - 6 (5 t - 6) t)}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.2

Factor $2$ out of $20 \cdot 3 t^{2} + 20 \cdot 4 - 6 (5 t - 6) t$ .

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Step 4.2.2.1

Factor $2$ out of $20 \cdot 3 t^{2}$ .

$\frac{{(5 t - 6)}^{3} (2 (10 \cdot 3 t^{2}) + 20 \cdot 4 - 6 (5 t - 6) t)}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.2.2

Factor $2$ out of $20 \cdot 4$ .

$\frac{{(5 t - 6)}^{3} (2 (10 \cdot 3 t^{2}) + 2 (10 \cdot 4) - 6 (5 t - 6) t)}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.2.3

Factor $2$ out of $- 6 (5 t - 6) t$ .

$\frac{{(5 t - 6)}^{3} (2 (10 \cdot 3 t^{2}) + 2 (10 \cdot 4) + 2 (- 3 (5 t - 6) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.2.4

Factor $2$ out of $2 (10 \cdot 3 t^{2}) + 2 (10 \cdot 4)$ .

$\frac{{(5 t - 6)}^{3} (2 (10 \cdot 3 t^{2} + 10 \cdot 4) + 2 (- 3 (5 t - 6) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.2.5

Factor $2$ out of $2 (10 \cdot 3 t^{2} + 10 \cdot 4) + 2 (- 3 (5 t - 6) t)$ .

$\frac{{(5 t - 6)}^{3} (2 (10 \cdot 3 t^{2} + 10 \cdot 4 - 3 (5 t - 6) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.3

Multiply $10$ by $3$ .

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 10 \cdot 4 - 3 (5 t - 6) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.4

Multiply $10$ by $4$ .

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 40 - 3 (5 t - 6) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.5

Apply the distributive property.

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 40 + (- 3 (5 t) - 3 \cdot - 6) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.6

Multiply $5$ by $- 3$ .

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 40 + (- 15 t - 3 \cdot - 6) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.7

Multiply $- 3$ by $- 6$ .

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 40 + (- 15 t + 18) t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.8

Apply the distributive property.

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 40 - 15 t \cdot t + 18 t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.9

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

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Step 4.2.9.1

Move $t$ .

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 40 - 15 (t \cdot t) + 18 t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.9.2

Multiply $t$ by $t$ .

$\frac{{(5 t - 6)}^{3} (2 (30 t^{2} + 40 - 15 t^{2} + 18 t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.10

Subtract $15 t^{2}$ from $30 t^{2}$ .

$\frac{{(5 t - 6)}^{3} (2 (15 t^{2} + 40 + 18 t))}{{(3 t^{2} + 4)}^{2}}$

Step 4.2.11

Reorder terms.

$\frac{{(5 t - 6)}^{3} \cdot 2 (15 t^{2} + 18 t + 40)}{{(3 t^{2} + 4)}^{2}}$

Step 4.3

Move $2$ to the left of ${(5 t - 6)}^{3}$ .

$\frac{2 {(5 t - 6)}^{3} (15 t^{2} + 18 t + 40)}{{(3 t^{2} + 4)}^{2}}$