Calculus Examples

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

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 $f (t) = 3 t - 1$ and $g (t) = t^{2} + 5 t$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 t - 1$ with respect to $t$ is $\frac{d}{d t} [3 t] + \frac{d}{d t} [- 1]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $1$ .

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

Step 2.5

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

$\frac{(t^{2} + 5 t) (3 + 0) - (3 t - 1) \frac{d}{d t} [t^{2} + 5 t]}{{(t^{2} + 5 t)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

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

Step 2.6.2

Move $3$ to the left of $t^{2} + 5 t$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Multiply $5$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $5$ by $3$ .

$\frac{3 t^{2} + 15 t + (- (3 t) - - 1) (2 t + 5)}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.2

Simplify each term.

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Step 3.3.1.2.1

Multiply $3$ by $- 1$ .

$\frac{3 t^{2} + 15 t + (- 3 t - - 1) (2 t + 5)}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.2.2

Multiply $- 1$ by $- 1$ .

$\frac{3 t^{2} + 15 t + (- 3 t + 1) (2 t + 5)}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.3

Expand $(- 3 t + 1) (2 t + 5)$ using the FOIL Method.

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Step 3.3.1.3.1

Apply the distributive property.

$\frac{3 t^{2} + 15 t - 3 t (2 t + 5) + 1 (2 t + 5)}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.3.2

Apply the distributive property.

$\frac{3 t^{2} + 15 t - 3 t (2 t) - 3 t \cdot 5 + 1 (2 t + 5)}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.3.3

Apply the distributive property.

$\frac{3 t^{2} + 15 t - 3 t (2 t) - 3 t \cdot 5 + 1 (2 t) + 1 \cdot 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.4

Simplify and combine like terms.

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Step 3.3.1.4.1

Simplify each term.

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Step 3.3.1.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 t^{2} + 15 t - 3 \cdot 2 t \cdot t - 3 t \cdot 5 + 1 (2 t) + 1 \cdot 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.4.1.2

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

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Step 3.3.1.4.1.2.1

Move $t$ .

$\frac{3 t^{2} + 15 t - 3 \cdot 2 (t \cdot t) - 3 t \cdot 5 + 1 (2 t) + 1 \cdot 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.4.1.2.2

Multiply $t$ by $t$ .

$\frac{3 t^{2} + 15 t - 3 \cdot 2 t^{2} - 3 t \cdot 5 + 1 (2 t) + 1 \cdot 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.4.1.3

Multiply $- 3$ by $2$ .

$\frac{3 t^{2} + 15 t - 6 t^{2} - 3 t \cdot 5 + 1 (2 t) + 1 \cdot 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.4.1.4

Multiply $5$ by $- 3$ .

$\frac{3 t^{2} + 15 t - 6 t^{2} - 15 t + 1 (2 t) + 1 \cdot 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.4.1.5

Multiply $2 t$ by $1$ .

$\frac{3 t^{2} + 15 t - 6 t^{2} - 15 t + 2 t + 1 \cdot 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.1.4.1.6

Multiply $5$ by $1$ .

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

Step 3.3.1.4.2

Add $- 15 t$ and $2 t$ .

$\frac{3 t^{2} + 15 t - 6 t^{2} - 13 t + 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.2

Subtract $6 t^{2}$ from $3 t^{2}$ .

$\frac{- 3 t^{2} + 15 t - 13 t + 5}{{(t^{2} + 5 t)}^{2}}$

Step 3.3.3

Subtract $13 t$ from $15 t$ .

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

Step 3.4

Factor by grouping.

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Step 3.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot 5 = - 15$ and whose sum is $b = 2$ .

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Step 3.4.1.1

Factor $2$ out of $2 t$ .

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

Step 3.4.1.2

Rewrite $2$ as $- 3$ plus $5$

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

Step 3.4.1.3

Apply the distributive property.

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

Step 3.4.2

Factor out the greatest common factor from each group.

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Step 3.4.2.1

Group the first two terms and the last two terms.

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

Step 3.4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.4.3

Factor the polynomial by factoring out the greatest common factor, $- t - 1$ .

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

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

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Step 3.5.1.1

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

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

Step 3.5.1.2

Factor $t$ out of $5 t$ .

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

Step 3.5.1.3

Factor $t$ out of $t \cdot t + t \cdot 5$ .

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

Step 3.5.2

Apply the product rule to $t (t + 5)$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Rewrite $- (t + 1)$ as $- 1 (t + 1)$ .

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

Step 3.10

Move the negative in front of the fraction.

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