Calculus Examples

$s = \frac{t^{2} + 5 t - 1}{t^{2}}$

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 $f (t) = t^{2} + 5 t - 1$ and $g (t) = t^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $t^{2} + 5 t - 1$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [5 t] + \frac{d}{d t} [- 1]$ .

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

Step 2.2

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

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

Step 3

Evaluate $\frac{d}{d t} [5 t]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $5$ by $1$ .

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

Step 4

Differentiate.

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Step 4.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$\frac{t^{2} (2 t + 5 + 0) - (t^{2} + 5 t - 1) \frac{d}{d t} [t^{2}]}{{(t^{2})}^{2}}$

Step 4.2

Add $2 t + 5$ and $0$ .

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

Step 4.3

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

$\frac{t^{2} (2 t + 5) - (t^{2} + 5 t - 1) (2 t)}{{(t^{2})}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{t^{2} (2 t) + t^{2} \cdot 5 - (t^{2} + 5 t - 1) (2 t)}{{(t^{2})}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{t^{2} (2 t) + t^{2} \cdot 5 + (- t^{2} - (5 t) - - 1) (2 t)}{{(t^{2})}^{2}}$

Step 5.3

Apply the distributive property.

$\frac{t^{2} (2 t) + t^{2} \cdot 5 - t^{2} (2 t) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 t^{2} t + t^{2} \cdot 5 - t^{2} (2 t) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.2

Multiply $t^{2}$ by $t$ by adding the exponents.

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Step 5.4.1.2.1

Move $t$ .

$\frac{2 (t \cdot t^{2}) + t^{2} \cdot 5 - t^{2} (2 t) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.2.2

Multiply $t$ by $t^{2}$ .

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Step 5.4.1.2.2.1

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

$\frac{2 (t^{1} t^{2}) + t^{2} \cdot 5 - t^{2} (2 t) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.2.2.2

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

$\frac{2 t^{1 + 2} + t^{2} \cdot 5 - t^{2} (2 t) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.2.3

Add $1$ and $2$ .

$\frac{2 t^{3} + t^{2} \cdot 5 - t^{2} (2 t) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.3

Move $5$ to the left of $t^{2}$ .

$\frac{2 t^{3} + 5 \cdot t^{2} - t^{2} (2 t) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.4

Rewrite using the commutative property of multiplication.

$\frac{2 t^{3} + 5 t^{2} - 1 \cdot 2 t^{2} t - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.5

Multiply $t^{2}$ by $t$ by adding the exponents.

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Step 5.4.1.5.1

Move $t$ .

$\frac{2 t^{3} + 5 t^{2} - 1 \cdot 2 (t \cdot t^{2}) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.5.2

Multiply $t$ by $t^{2}$ .

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Step 5.4.1.5.2.1

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

$\frac{2 t^{3} + 5 t^{2} - 1 \cdot 2 (t^{1} t^{2}) - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.5.2.2

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

$\frac{2 t^{3} + 5 t^{2} - 1 \cdot 2 t^{1 + 2} - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.5.3

Add $1$ and $2$ .

$\frac{2 t^{3} + 5 t^{2} - 1 \cdot 2 t^{3} - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.6

Multiply $- 1$ by $2$ .

$\frac{2 t^{3} + 5 t^{2} - 2 t^{3} - (5 t) (2 t) - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.7

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

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Step 5.4.1.7.1

Move $t$ .

$\frac{2 t^{3} + 5 t^{2} - 2 t^{3} - (5 (t \cdot t)) \cdot 2 - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.7.2

Multiply $t$ by $t$ .

$\frac{2 t^{3} + 5 t^{2} - 2 t^{3} - (5 t^{2}) \cdot 2 - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.8

Multiply $5$ by $- 1$ .

$\frac{2 t^{3} + 5 t^{2} - 2 t^{3} - 5 t^{2} \cdot 2 - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.9

Multiply $2$ by $- 5$ .

$\frac{2 t^{3} + 5 t^{2} - 2 t^{3} - 10 t^{2} - - 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.10

Multiply $- 1$ by $- 1$ .

$\frac{2 t^{3} + 5 t^{2} - 2 t^{3} - 10 t^{2} + 1 (2 t)}{{(t^{2})}^{2}}$

Step 5.4.1.11

Multiply $2 t$ by $1$ .

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

Step 5.4.2

Combine the opposite terms in $2 t^{3} + 5 t^{2} - 2 t^{3} - 10 t^{2} + 2 t$ .

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Step 5.4.2.1

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

$\frac{5 t^{2} + 0 - 10 t^{2} + 2 t}{{(t^{2})}^{2}}$

Step 5.4.2.2

Add $5 t^{2}$ and $0$ .

$\frac{5 t^{2} - 10 t^{2} + 2 t}{{(t^{2})}^{2}}$

Step 5.4.3

Subtract $10 t^{2}$ from $5 t^{2}$ .

$\frac{- 5 t^{2} + 2 t}{{(t^{2})}^{2}}$

Step 5.5

Combine terms.

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Step 5.5.1

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

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Step 5.5.1.1

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

$\frac{- 5 t^{2} + 2 t}{t^{2 \cdot 2}}$

Step 5.5.1.2

Multiply $2$ by $2$ .

$\frac{- 5 t^{2} + 2 t}{t^{4}}$

Step 5.5.2

Factor $t$ out of $- 5 t^{2} + 2 t$ .

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Step 5.5.2.1

Factor $t$ out of $- 5 t^{2}$ .

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

Step 5.5.2.2

Factor $t$ out of $2 t$ .

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

Step 5.5.2.3

Factor $t$ out of $t (- 5 t) + t \cdot 2$ .

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

Step 5.5.3

Cancel the common factors.

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Step 5.5.3.1

Factor $t$ out of $t^{4}$ .

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

Step 5.5.3.2

Cancel the common factor.

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

Step 5.5.3.3

Rewrite the expression.

$\frac{- 5 t + 2}{t^{3}}$

Step 5.6

Factor $- 1$ out of $- 5 t$ .

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

Step 5.7

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (5 t) - 1 (- 2)}{t^{3}}$

Step 5.8

Factor $- 1$ out of $- (5 t) - 1 (- 2)$ .

$\frac{- (5 t - 2)}{t^{3}}$

Step 5.9

Rewrite $- (5 t - 2)$ as $- 1 (5 t - 2)$ .

$\frac{- 1 (5 t - 2)}{t^{3}}$

Step 5.10

Move the negative in front of the fraction.

$- \frac{5 t - 2}{t^{3}}$