Calculus Examples

$f (x) = 50 - \frac{25 t^{2}}{{(t + 3)}^{2}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $50 - \frac{25 t^{2}}{{(t + 3)}^{2}}$ with respect to $t$ is $\frac{d}{d t} [50] + \frac{d}{d t} [- \frac{25 t^{2}}{{(t + 3)}^{2}}]$ .

$\frac{d}{d t} [50] + \frac{d}{d t} [- \frac{25 t^{2}}{{(t + 3)}^{2}}]$

Step 1.2

Since $50$ is constant with respect to $t$ , the derivative of $50$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [- \frac{25 t^{2}}{{(t + 3)}^{2}}]$

Step 2

Evaluate $\frac{d}{d t} [- \frac{25 t^{2}}{{(t + 3)}^{2}}]$ .

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Step 2.1

Since $- 25$ is constant with respect to $t$ , the derivative of $- \frac{25 t^{2}}{{(t + 3)}^{2}}$ with respect to $t$ is $- 25 \frac{d}{d t} [\frac{t^{2}}{{(t + 3)}^{2}}]$ .

$0 - 25 \frac{d}{d t} [\frac{t^{2}}{{(t + 3)}^{2}}]$

Step 2.2

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

$0 - 25 \frac{{(t + 3)}^{2} \frac{d}{d t} [t^{2}] - t^{2} \frac{d}{d t} [{(t + 3)}^{2}]}{{({(t + 3)}^{2})}^{2}}$

Step 2.3

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

$0 - 25 \frac{{(t + 3)}^{2} (2 t) - t^{2} \frac{d}{d t} [{(t + 3)}^{2}]}{{({(t + 3)}^{2})}^{2}}$

Step 2.4

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 $g (t) = t + 3$ .

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Step 2.4.1

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

$0 - 25 \frac{{(t + 3)}^{2} (2 t) - t^{2} (\frac{d}{d u} [u^{2}] \frac{d}{d t} [t + 3])}{{({(t + 3)}^{2})}^{2}}$

Step 2.4.2

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

$0 - 25 \frac{{(t + 3)}^{2} (2 t) - t^{2} (2 u \frac{d}{d t} [t + 3])}{{({(t + 3)}^{2})}^{2}}$

Step 2.4.3

Replace all occurrences of $u$ with $t + 3$ .

$0 - 25 \frac{{(t + 3)}^{2} (2 t) - t^{2} (2 (t + 3) \frac{d}{d t} [t + 3])}{{({(t + 3)}^{2})}^{2}}$

Step 2.5

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

$0 - 25 \frac{{(t + 3)}^{2} (2 t) - t^{2} (2 (t + 3) (\frac{d}{d t} [t] + \frac{d}{d t} [3]))}{{({(t + 3)}^{2})}^{2}}$

Step 2.6

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

$0 - 25 \frac{{(t + 3)}^{2} (2 t) - t^{2} (2 (t + 3) (1 + \frac{d}{d t} [3]))}{{({(t + 3)}^{2})}^{2}}$

Step 2.7

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

$0 - 25 \frac{{(t + 3)}^{2} (2 t) - t^{2} (2 (t + 3) (1 + 0))}{{({(t + 3)}^{2})}^{2}}$

Step 2.8

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

$0 - 25 \frac{2 \cdot {(t + 3)}^{2} t - t^{2} (2 (t + 3) (1 + 0))}{{({(t + 3)}^{2})}^{2}}$

Step 2.9

Add $1$ and $0$ .

$0 - 25 \frac{2 {(t + 3)}^{2} t - t^{2} (2 (t + 3) \cdot 1)}{{({(t + 3)}^{2})}^{2}}$

Step 2.10

Multiply $2$ by $1$ .

$0 - 25 \frac{2 {(t + 3)}^{2} t - t^{2} (2 (t + 3))}{{({(t + 3)}^{2})}^{2}}$

Step 2.11

Multiply $2$ by $- 1$ .

$0 - 25 \frac{2 {(t + 3)}^{2} t - 2 t^{2} (t + 3)}{{({(t + 3)}^{2})}^{2}}$

Step 2.12

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

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Step 2.12.1

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

$0 - 25 \frac{2 {(t + 3)}^{2} t - 2 t^{2} (t + 3)}{{(t + 3)}^{2 \cdot 2}}$

Step 2.12.2

Multiply $2$ by $2$ .

$0 - 25 \frac{2 {(t + 3)}^{2} t - 2 t^{2} (t + 3)}{{(t + 3)}^{4}}$

Step 2.13

Combine $- 25$ and $\frac{2 {(t + 3)}^{2} t - 2 t^{2} (t + 3)}{{(t + 3)}^{4}}$ .

$0 + \frac{- 25 (2 {(t + 3)}^{2} t - 2 t^{2} (t + 3))}{{(t + 3)}^{4}}$

Step 2.14

Move the negative in front of the fraction.

$0 - \frac{25 (2 {(t + 3)}^{2} t - 2 t^{2} (t + 3))}{{(t + 3)}^{4}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$0 - \frac{25 (2 {(t + 3)}^{2} t - 2 t^{2} t - 2 t^{2} \cdot 3)}{{(t + 3)}^{4}}$

Step 3.2

Apply the distributive property.

$0 - \frac{25 (2 {(t + 3)}^{2} t) + 25 (- 2 t^{2} t) + 25 (- 2 t^{2} \cdot 3)}{{(t + 3)}^{4}}$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $2$ by $25$ .

$0 - \frac{50 ({(t + 3)}^{2} t) + 25 (- 2 t^{2} t) + 25 (- 2 t^{2} \cdot 3)}{{(t + 3)}^{4}}$

Step 3.3.2

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

$0 - \frac{50 {(t + 3)}^{2} t + 25 (- 2 (t^{1} t^{2})) + 25 (- 2 t^{2} \cdot 3)}{{(t + 3)}^{4}}$

Step 3.3.3

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

$0 - \frac{50 {(t + 3)}^{2} t + 25 (- 2 t^{1 + 2}) + 25 (- 2 t^{2} \cdot 3)}{{(t + 3)}^{4}}$

Step 3.3.4

Add $1$ and $2$ .

$0 - \frac{50 {(t + 3)}^{2} t + 25 (- 2 t^{3}) + 25 (- 2 t^{2} \cdot 3)}{{(t + 3)}^{4}}$

Step 3.3.5

Multiply $- 2$ by $25$ .

$0 - \frac{50 {(t + 3)}^{2} t - 50 t^{3} + 25 (- 2 t^{2} \cdot 3)}{{(t + 3)}^{4}}$

Step 3.3.6

Multiply $3$ by $- 2$ .

$0 - \frac{50 {(t + 3)}^{2} t - 50 t^{3} + 25 (- 6 t^{2})}{{(t + 3)}^{4}}$

Step 3.3.7

Multiply $- 6$ by $25$ .

$0 - \frac{50 {(t + 3)}^{2} t - 50 t^{3} - 150 t^{2}}{{(t + 3)}^{4}}$

Step 3.3.8

Subtract $\frac{50 {(t + 3)}^{2} t - 50 t^{3} - 150 t^{2}}{{(t + 3)}^{4}}$ from $0$ .

$- \frac{50 {(t + 3)}^{2} t - 50 t^{3} - 150 t^{2}}{{(t + 3)}^{4}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $50 t$ out of $50 {(t + 3)}^{2} t - 50 t^{3} - 150 t^{2}$ .

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Step 3.4.1.1

Factor $50 t$ out of $50 {(t + 3)}^{2} t$ .

$- \frac{50 t ({(t + 3)}^{2}) - 50 t^{3} - 150 t^{2}}{{(t + 3)}^{4}}$

Step 3.4.1.2

Factor $50 t$ out of $- 50 t^{3}$ .

$- \frac{50 t ({(t + 3)}^{2}) + 50 t (- t^{2}) - 150 t^{2}}{{(t + 3)}^{4}}$

Step 3.4.1.3

Factor $50 t$ out of $- 150 t^{2}$ .

$- \frac{50 t ({(t + 3)}^{2}) + 50 t (- t^{2}) + 50 t (- 3 t)}{{(t + 3)}^{4}}$

Step 3.4.1.4

Factor $50 t$ out of $50 t ({(t + 3)}^{2}) + 50 t (- t^{2})$ .

$- \frac{50 t ({(t + 3)}^{2} - t^{2}) + 50 t (- 3 t)}{{(t + 3)}^{4}}$

Step 3.4.1.5

Factor $50 t$ out of $50 t ({(t + 3)}^{2} - t^{2}) + 50 t (- 3 t)$ .

$- \frac{50 t ({(t + 3)}^{2} - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.2

Rewrite ${(t + 3)}^{2}$ as $(t + 3) (t + 3)$ .

$- \frac{50 t ((t + 3) (t + 3) - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.3

Expand $(t + 3) (t + 3)$ using the FOIL Method.

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Step 3.4.3.1

Apply the distributive property.

$- \frac{50 t (t (t + 3) + 3 (t + 3) - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.3.2

Apply the distributive property.

$- \frac{50 t (t \cdot t + t \cdot 3 + 3 (t + 3) - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.3.3

Apply the distributive property.

$- \frac{50 t (t \cdot t + t \cdot 3 + 3 t + 3 \cdot 3 - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.4

Simplify and combine like terms.

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Step 3.4.4.1

Simplify each term.

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Step 3.4.4.1.1

Multiply $t$ by $t$ .

$- \frac{50 t (t^{2} + t \cdot 3 + 3 t + 3 \cdot 3 - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.4.1.2

Move $3$ to the left of $t$ .

$- \frac{50 t (t^{2} + 3 \cdot t + 3 t + 3 \cdot 3 - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.4.1.3

Multiply $3$ by $3$ .

$- \frac{50 t (t^{2} + 3 t + 3 t + 9 - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.4.2

Add $3 t$ and $3 t$ .

$- \frac{50 t (t^{2} + 6 t + 9 - t^{2} - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.5

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

$- \frac{50 t (6 t + 9 + 0 - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.6

Add $6 t$ and $0$ .

$- \frac{50 t (6 t + 9 - 3 t)}{{(t + 3)}^{4}}$

Step 3.4.7

Subtract $3 t$ from $6 t$ .

$- \frac{50 t (3 t + 9)}{{(t + 3)}^{4}}$

Step 3.4.8

Factor $3$ out of $3 t + 9$ .

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Step 3.4.8.1

Factor $3$ out of $3 t$ .

$- \frac{50 t (3 (t) + 9)}{{(t + 3)}^{4}}$

Step 3.4.8.2

Factor $3$ out of $9$ .

$- \frac{50 t (3 t + 3 \cdot 3)}{{(t + 3)}^{4}}$

Step 3.4.8.3

Factor $3$ out of $3 t + 3 \cdot 3$ .

$- \frac{50 t (3 (t + 3))}{{(t + 3)}^{4}}$

$- \frac{50 t \cdot 3 (t + 3)}{{(t + 3)}^{4}}$

Step 3.4.9

Multiply $3$ by $50$ .

$- \frac{150 t (t + 3)}{{(t + 3)}^{4}}$

Step 3.5

Cancel the common factor of $t + 3$ and ${(t + 3)}^{4}$ .

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Step 3.5.1

Factor $t + 3$ out of $150 t (t + 3)$ .

$- \frac{(t + 3) (150 t)}{{(t + 3)}^{4}}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

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

$- \frac{(t + 3) (150 t)}{(t + 3) {(t + 3)}^{3}}$

Step 3.5.2.2

Cancel the common factor.

$- \frac{(t + 3) (150 t)}{(t + 3) {(t + 3)}^{3}}$

Step 3.5.2.3

Rewrite the expression.

$- \frac{150 t}{{(t + 3)}^{3}}$