Calculus Examples

$y = \frac{t^{2} + 9}{{(8 t - 3)}^{6}}$

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 $f (t) = t^{2} + 9$ and $g (t) = {(8 t - 3)}^{6}$ .

$\frac{{(8 t - 3)}^{6} \frac{d}{d t} [t^{2} + 9] - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{({(8 t - 3)}^{6})}^{2}}$

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${({(8 t - 3)}^{6})}^{2}$ .

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Step 2.1.1

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

$\frac{{(8 t - 3)}^{6} \frac{d}{d t} [t^{2} + 9] - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{(8 t - 3)}^{6 \cdot 2}}$

Step 2.1.2

Multiply $6$ by $2$ .

$\frac{{(8 t - 3)}^{6} \frac{d}{d t} [t^{2} + 9] - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{(8 t - 3)}^{12}}$

Step 2.2

By the Sum Rule, the derivative of $t^{2} + 9$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [9]$ .

$\frac{{(8 t - 3)}^{6} (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [9]) - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{(8 t - 3)}^{12}}$

Step 2.3

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

$\frac{{(8 t - 3)}^{6} (2 t + \frac{d}{d t} [9]) - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{(8 t - 3)}^{12}}$

Step 2.4

Since $9$ is constant with respect to $t$ , the derivative of $9$ with respect to $t$ is $0$ .

$\frac{{(8 t - 3)}^{6} (2 t + 0) - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{(8 t - 3)}^{12}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Add $2 t$ and $0$ .

$\frac{{(8 t - 3)}^{6} (2 t) - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{(8 t - 3)}^{12}}$

Step 2.5.2

Move $2$ to the left of ${(8 t - 3)}^{6}$ .

$\frac{2 \cdot {(8 t - 3)}^{6} t - (t^{2} + 9) \frac{d}{d t} [{(8 t - 3)}^{6}]}{{(8 t - 3)}^{12}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{6}$ and $g (t) = 8 t - 3$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $8 t - 3$ .

$\frac{2 {(8 t - 3)}^{6} t - (t^{2} + 9) (\frac{d}{d u} [u^{6}] \frac{d}{d t} [8 t - 3])}{{(8 t - 3)}^{12}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$\frac{2 {(8 t - 3)}^{6} t - (t^{2} + 9) (6 u^{5} \frac{d}{d t} [8 t - 3])}{{(8 t - 3)}^{12}}$

Step 3.3

Replace all occurrences of $u$ with $8 t - 3$ .

$\frac{2 {(8 t - 3)}^{6} t - (t^{2} + 9) (6 {(8 t - 3)}^{5} \frac{d}{d t} [8 t - 3])}{{(8 t - 3)}^{12}}$

Step 4

Differentiate.

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Step 4.1

Multiply $6$ by $- 1$ .

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) ({(8 t - 3)}^{5} \frac{d}{d t} [8 t - 3])}{{(8 t - 3)}^{12}}$

Step 4.2

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

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) ({(8 t - 3)}^{5} (\frac{d}{d t} [8 t] + \frac{d}{d t} [- 3]))}{{(8 t - 3)}^{12}}$

Step 4.3

Since $8$ is constant with respect to $t$ , the derivative of $8 t$ with respect to $t$ is $8 \frac{d}{d t} [t]$ .

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) ({(8 t - 3)}^{5} (8 \frac{d}{d t} [t] + \frac{d}{d t} [- 3]))}{{(8 t - 3)}^{12}}$

Step 4.4

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

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) ({(8 t - 3)}^{5} (8 \cdot 1 + \frac{d}{d t} [- 3]))}{{(8 t - 3)}^{12}}$

Step 4.5

Multiply $8$ by $1$ .

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) ({(8 t - 3)}^{5} (8 + \frac{d}{d t} [- 3]))}{{(8 t - 3)}^{12}}$

Step 4.6

Since $- 3$ is constant with respect to $t$ , the derivative of $- 3$ with respect to $t$ is $0$ .

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) ({(8 t - 3)}^{5} (8 + 0))}{{(8 t - 3)}^{12}}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Add $8$ and $0$ .

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) ({(8 t - 3)}^{5} \cdot 8)}{{(8 t - 3)}^{12}}$

Step 4.7.2

Move $8$ to the left of ${(8 t - 3)}^{5}$ .

$\frac{2 {(8 t - 3)}^{6} t - 6 (t^{2} + 9) (8 \cdot {(8 t - 3)}^{5})}{{(8 t - 3)}^{12}}$

Step 4.7.3

Multiply $8$ by $- 6$ .

$\frac{2 {(8 t - 3)}^{6} t - 48 (t^{2} + 9) {(8 t - 3)}^{5}}{{(8 t - 3)}^{12}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{2 {(8 t - 3)}^{6} t + (- 48 t^{2} - 48 \cdot 9) {(8 t - 3)}^{5}}{{(8 t - 3)}^{12}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Factor ${(8 t - 3)}^{5}$ out of $2 {(8 t - 3)}^{6} t + (- 48 t^{2} - 48 \cdot 9) {(8 t - 3)}^{5}$ .

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Step 5.2.1.1

Factor ${(8 t - 3)}^{5}$ out of $2 {(8 t - 3)}^{6} t$ .

$\frac{{(8 t - 3)}^{5} (2 (8 t - 3) t) + (- 48 t^{2} - 48 \cdot 9) {(8 t - 3)}^{5}}{{(8 t - 3)}^{12}}$

Step 5.2.1.2

Factor ${(8 t - 3)}^{5}$ out of $(- 48 t^{2} - 48 \cdot 9) {(8 t - 3)}^{5}$ .

$\frac{{(8 t - 3)}^{5} (2 (8 t - 3) t) + {(8 t - 3)}^{5} (- 48 t^{2} - 48 \cdot 9)}{{(8 t - 3)}^{12}}$

Step 5.2.1.3

Factor ${(8 t - 3)}^{5}$ out of ${(8 t - 3)}^{5} (2 (8 t - 3) t) + {(8 t - 3)}^{5} (- 48 t^{2} - 48 \cdot 9)$ .

$\frac{{(8 t - 3)}^{5} (2 (8 t - 3) t - 48 t^{2} - 48 \cdot 9)}{{(8 t - 3)}^{12}}$

Step 5.2.2

Factor $2$ out of $2 (8 t - 3) t - 48 t^{2} - 48 \cdot 9$ .

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Step 5.2.2.1

Factor $2$ out of $2 (8 t - 3) t$ .

$\frac{{(8 t - 3)}^{5} (2 ((8 t - 3) t) - 48 t^{2} - 48 \cdot 9)}{{(8 t - 3)}^{12}}$

Step 5.2.2.2

Factor $2$ out of $- 48 t^{2}$ .

$\frac{{(8 t - 3)}^{5} (2 ((8 t - 3) t) + 2 (- 24 t^{2}) - 48 \cdot 9)}{{(8 t - 3)}^{12}}$

Step 5.2.2.3

Factor $2$ out of $- 48 \cdot 9$ .

$\frac{{(8 t - 3)}^{5} (2 ((8 t - 3) t) + 2 (- 24 t^{2}) + 2 (- 24 \cdot 9))}{{(8 t - 3)}^{12}}$

Step 5.2.2.4

Factor $2$ out of $2 ((8 t - 3) t) + 2 (- 24 t^{2})$ .

$\frac{{(8 t - 3)}^{5} (2 ((8 t - 3) t - 24 t^{2}) + 2 (- 24 \cdot 9))}{{(8 t - 3)}^{12}}$

Step 5.2.2.5

Factor $2$ out of $2 ((8 t - 3) t - 24 t^{2}) + 2 (- 24 \cdot 9)$ .

$\frac{{(8 t - 3)}^{5} (2 ((8 t - 3) t - 24 t^{2} - 24 \cdot 9))}{{(8 t - 3)}^{12}}$

Step 5.2.3

Apply the distributive property.

$\frac{{(8 t - 3)}^{5} (2 (8 t \cdot t - 3 t - 24 t^{2} - 24 \cdot 9))}{{(8 t - 3)}^{12}}$

Step 5.2.4

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

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Step 5.2.4.1

Move $t$ .

$\frac{{(8 t - 3)}^{5} (2 (8 (t \cdot t) - 3 t - 24 t^{2} - 24 \cdot 9))}{{(8 t - 3)}^{12}}$

Step 5.2.4.2

Multiply $t$ by $t$ .

$\frac{{(8 t - 3)}^{5} (2 (8 t^{2} - 3 t - 24 t^{2} - 24 \cdot 9))}{{(8 t - 3)}^{12}}$

Step 5.2.5

Multiply $- 24$ by $9$ .

$\frac{{(8 t - 3)}^{5} (2 (8 t^{2} - 3 t - 24 t^{2} - 216))}{{(8 t - 3)}^{12}}$

Step 5.2.6

Subtract $24 t^{2}$ from $8 t^{2}$ .

$\frac{{(8 t - 3)}^{5} \cdot 2 (- 16 t^{2} - 3 t - 216)}{{(8 t - 3)}^{12}}$

Step 5.3

Combine terms.

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Step 5.3.1

Move $2$ to the left of ${(8 t - 3)}^{5}$ .

$\frac{2 \cdot {(8 t - 3)}^{5} (- 16 t^{2} - 3 t - 216)}{{(8 t - 3)}^{12}}$

Step 5.3.2

Cancel the common factor of ${(8 t - 3)}^{5}$ and ${(8 t - 3)}^{12}$ .

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Step 5.3.2.1

Factor ${(8 t - 3)}^{5}$ out of $2 {(8 t - 3)}^{5} (- 16 t^{2} - 3 t - 216)$ .

$\frac{{(8 t - 3)}^{5} (2 (- 16 t^{2} - 3 t - 216))}{{(8 t - 3)}^{12}}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor ${(8 t - 3)}^{5}$ out of ${(8 t - 3)}^{12}$ .

$\frac{{(8 t - 3)}^{5} (2 (- 16 t^{2} - 3 t - 216))}{{(8 t - 3)}^{5} {(8 t - 3)}^{7}}$

Step 5.3.2.2.2

Cancel the common factor.

$\frac{{(8 t - 3)}^{5} (2 (- 16 t^{2} - 3 t - 216))}{{(8 t - 3)}^{5} {(8 t - 3)}^{7}}$

Step 5.3.2.2.3

Rewrite the expression.

$\frac{2 (- 16 t^{2} - 3 t - 216)}{{(8 t - 3)}^{7}}$

Step 5.4

Factor $- 1$ out of $- 16 t^{2}$ .

$\frac{2 (- (16 t^{2}) - 3 t - 216)}{{(8 t - 3)}^{7}}$

Step 5.5

Factor $- 1$ out of $- 3 t$ .

$\frac{2 (- (16 t^{2}) - (3 t) - 216)}{{(8 t - 3)}^{7}}$

Step 5.6

Factor $- 1$ out of $- (16 t^{2}) - (3 t)$ .

$\frac{2 (- (16 t^{2} + 3 t) - 216)}{{(8 t - 3)}^{7}}$

Step 5.7

Rewrite $- 216$ as $- 1 (216)$ .

$\frac{2 (- (16 t^{2} + 3 t) - 1 (216))}{{(8 t - 3)}^{7}}$

Step 5.8

Factor $- 1$ out of $- (16 t^{2} + 3 t) - 1 (216)$ .

$\frac{2 (- (16 t^{2} + 3 t + 216))}{{(8 t - 3)}^{7}}$

Step 5.9

Rewrite $- (16 t^{2} + 3 t + 216)$ as $- 1 (16 t^{2} + 3 t + 216)$ .

$\frac{2 (- 1 (16 t^{2} + 3 t + 216))}{{(8 t - 3)}^{7}}$

Step 5.10

Move the negative in front of the fraction.

$- \frac{2 (16 t^{2} + 3 t + 216)}{{(8 t - 3)}^{7}}$