Calculus Examples

$p (t) = 20 - \frac{7}{t + 2}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $20 - \frac{7}{t + 2}$ with respect to $t$ is $\frac{d}{d t} [20] + \frac{d}{d t} [- \frac{7}{t + 2}]$ .

$\frac{d}{d t} [20] + \frac{d}{d t} [- \frac{7}{t + 2}]$

Step 1.2

Since $20$ is constant with respect to $t$ , the derivative of $20$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [- \frac{7}{t + 2}]$

Step 2

Evaluate $\frac{d}{d t} [- \frac{7}{t + 2}]$ .

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Step 2.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- \frac{7}{t + 2}$ with respect to $t$ is $- 7 \frac{d}{d t} [\frac{1}{t + 2}]$ .

$0 - 7 \frac{d}{d t} [\frac{1}{t + 2}]$

Step 2.2

Rewrite $\frac{1}{t + 2}$ as ${(t + 2)}^{- 1}$ .

$0 - 7 \frac{d}{d t} [{(t + 2)}^{- 1}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t + 2$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $t + 2$ .

$0 - 7 (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [t + 2])$

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $t + 2$ .

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

Step 2.4

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

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

Step 2.5

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

$0 - 7 (- {(t + 2)}^{- 2} (1 + \frac{d}{d t} [2]))$

Step 2.6

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

$0 - 7 (- {(t + 2)}^{- 2} (1 + 0))$

Step 2.7

Add $1$ and $0$ .

$0 - 7 (- {(t + 2)}^{- 2} \cdot 1)$

Step 2.8

Multiply $- 1$ by $1$ .

$0 - 7 (- {(t + 2)}^{- 2})$

Step 2.9

Multiply $- 1$ by $- 7$ .

$0 + 7 {(t + 2)}^{- 2}$

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + 7 \frac{1}{{(t + 2)}^{2}}$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $7$ and $\frac{1}{{(t + 2)}^{2}}$ .

$0 + \frac{7}{{(t + 2)}^{2}}$

Step 3.2.2

Add $0$ and $\frac{7}{{(t + 2)}^{2}}$ .

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