Calculus Examples

$y = \frac{2 t - 3}{t + 2}$

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 $f (t) = 2 t - 3$ and $g (t) = t + 2$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $1$ .

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $2$ and $0$ .

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

Step 2.6.2

Move $2$ to the left of $t + 2$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

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

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $1$ and $0$ .

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

Step 2.10.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

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 $2$ by $2$ .

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

Step 3.3.1.2

Multiply $2$ by $- 1$ .

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

Step 3.3.1.3

Multiply $- 1$ by $- 3$ .

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

Step 3.3.2

Combine the opposite terms in $2 t + 4 - 2 t + 3$ .

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Step 3.3.2.1

Subtract $2 t$ from $2 t$ .

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

Step 3.3.2.2

Add $0$ and $4$ .

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

Step 3.3.3

Add $4$ and $3$ .

$\frac{7}{{(t + 2)}^{2}}$