Calculus Examples

$y = \frac{3 t - 7}{t^{2} + 5 t - 4}$

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 $f (t) = 3 t - 7$ and $g (t) = t^{2} + 5 t - 4$ .

$\frac{(t^{2} + 5 t - 4) \frac{d}{d t} [3 t - 7] - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 t - 7$ with respect to $t$ is $\frac{d}{d t} [3 t] + \frac{d}{d t} [- 7]$ .

$\frac{(t^{2} + 5 t - 4) (\frac{d}{d t} [3 t] + \frac{d}{d t} [- 7]) - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.2

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

$\frac{(t^{2} + 5 t - 4) (3 \frac{d}{d t} [t] + \frac{d}{d t} [- 7]) - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.3

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

$\frac{(t^{2} + 5 t - 4) (3 \cdot 1 + \frac{d}{d t} [- 7]) - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.4

Multiply $3$ by $1$ .

$\frac{(t^{2} + 5 t - 4) (3 + \frac{d}{d t} [- 7]) - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.5

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7$ with respect to $t$ is $0$ .

$\frac{(t^{2} + 5 t - 4) (3 + 0) - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3$ and $0$ .

$\frac{(t^{2} + 5 t - 4) \cdot 3 - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.6.2

Move $3$ to the left of $t^{2} + 5 t - 4$ .

$\frac{3 \cdot (t^{2} + 5 t - 4) - (3 t - 7) \frac{d}{d t} [t^{2} + 5 t - 4]}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $t^{2} + 5 t - 4$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [5 t] + \frac{d}{d t} [- 4]$ .

$\frac{3 (t^{2} + 5 t - 4) - (3 t - 7) (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [5 t] + \frac{d}{d t} [- 4])}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.8

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

$\frac{3 (t^{2} + 5 t - 4) - (3 t - 7) (2 t + \frac{d}{d t} [5 t] + \frac{d}{d t} [- 4])}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.9

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

$\frac{3 (t^{2} + 5 t - 4) - (3 t - 7) (2 t + 5 \frac{d}{d t} [t] + \frac{d}{d t} [- 4])}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.10

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

$\frac{3 (t^{2} + 5 t - 4) - (3 t - 7) (2 t + 5 \cdot 1 + \frac{d}{d t} [- 4])}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.11

Multiply $5$ by $1$ .

$\frac{3 (t^{2} + 5 t - 4) - (3 t - 7) (2 t + 5 + \frac{d}{d t} [- 4])}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.12

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

$\frac{3 (t^{2} + 5 t - 4) - (3 t - 7) (2 t + 5 + 0)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 2.13

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

$\frac{3 (t^{2} + 5 t - 4) - (3 t - 7) (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{3 t^{2} + 3 (5 t) + 3 \cdot - 4 - (3 t - 7) (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{3 t^{2} + 3 (5 t) + 3 \cdot - 4 + (- (3 t) - - 7) (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

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 $3$ .

$\frac{3 t^{2} + 15 t + 3 \cdot - 4 + (- (3 t) - - 7) (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.2

Multiply $3$ by $- 4$ .

$\frac{3 t^{2} + 15 t - 12 + (- (3 t) - - 7) (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.3

Simplify each term.

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Step 3.3.1.3.1

Multiply $3$ by $- 1$ .

$\frac{3 t^{2} + 15 t - 12 + (- 3 t - - 7) (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.3.2

Multiply $- 1$ by $- 7$ .

$\frac{3 t^{2} + 15 t - 12 + (- 3 t + 7) (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.4

Expand $(- 3 t + 7) (2 t + 5)$ using the FOIL Method.

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Step 3.3.1.4.1

Apply the distributive property.

$\frac{3 t^{2} + 15 t - 12 - 3 t (2 t + 5) + 7 (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.4.2

Apply the distributive property.

$\frac{3 t^{2} + 15 t - 12 - 3 t (2 t) - 3 t \cdot 5 + 7 (2 t + 5)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.4.3

Apply the distributive property.

$\frac{3 t^{2} + 15 t - 12 - 3 t (2 t) - 3 t \cdot 5 + 7 (2 t) + 7 \cdot 5}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5

Simplify and combine like terms.

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Step 3.3.1.5.1

Simplify each term.

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Step 3.3.1.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 t^{2} + 15 t - 12 - 3 \cdot 2 t \cdot t - 3 t \cdot 5 + 7 (2 t) + 7 \cdot 5}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5.1.2

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

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Step 3.3.1.5.1.2.1

Move $t$ .

$\frac{3 t^{2} + 15 t - 12 - 3 \cdot 2 (t \cdot t) - 3 t \cdot 5 + 7 (2 t) + 7 \cdot 5}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5.1.2.2

Multiply $t$ by $t$ .

$\frac{3 t^{2} + 15 t - 12 - 3 \cdot 2 t^{2} - 3 t \cdot 5 + 7 (2 t) + 7 \cdot 5}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5.1.3

Multiply $- 3$ by $2$ .

$\frac{3 t^{2} + 15 t - 12 - 6 t^{2} - 3 t \cdot 5 + 7 (2 t) + 7 \cdot 5}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5.1.4

Multiply $5$ by $- 3$ .

$\frac{3 t^{2} + 15 t - 12 - 6 t^{2} - 15 t + 7 (2 t) + 7 \cdot 5}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5.1.5

Multiply $2$ by $7$ .

$\frac{3 t^{2} + 15 t - 12 - 6 t^{2} - 15 t + 14 t + 7 \cdot 5}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5.1.6

Multiply $7$ by $5$ .

$\frac{3 t^{2} + 15 t - 12 - 6 t^{2} - 15 t + 14 t + 35}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.1.5.2

Add $- 15 t$ and $14 t$ .

$\frac{3 t^{2} + 15 t - 12 - 6 t^{2} - t + 35}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.2

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

$\frac{- 3 t^{2} + 15 t - 12 - t + 35}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.3

Subtract $t$ from $15 t$ .

$\frac{- 3 t^{2} + 14 t - 12 + 35}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.3.4

Add $- 12$ and $35$ .

$\frac{- 3 t^{2} + 14 t + 23}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.4

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

$\frac{- (3 t^{2}) + 14 t + 23}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.5

Factor $- 1$ out of $14 t$ .

$\frac{- (3 t^{2}) - (- 14 t) + 23}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.6

Factor $- 1$ out of $- (3 t^{2}) - (- 14 t)$ .

$\frac{- (3 t^{2} - 14 t) + 23}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.7

Rewrite $23$ as $- 1 (- 23)$ .

$\frac{- (3 t^{2} - 14 t) - 1 (- 23)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.8

Factor $- 1$ out of $- (3 t^{2} - 14 t) - 1 (- 23)$ .

$\frac{- (3 t^{2} - 14 t - 23)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.9

Rewrite $- (3 t^{2} - 14 t - 23)$ as $- 1 (3 t^{2} - 14 t - 23)$ .

$\frac{- 1 (3 t^{2} - 14 t - 23)}{{(t^{2} + 5 t - 4)}^{2}}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{3 t^{2} - 14 t - 23}{{(t^{2} + 5 t - 4)}^{2}}$