Calculus Examples

$\frac{n}{n + 7}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d n} [\frac{f (n)}{g (n)}]$ is $\frac{g (n) \frac{d}{d n} [f (n)] - f (n) \frac{d}{d n} [g (n)]}{{g (n)}^{2}}$ where $f (n) = n$ and $g (n) = n + 7$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Multiply $n + 7$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify by adding terms.

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Step 2.6.1

Add $1$ and $0$ .

$\frac{n + 7 - n \cdot 1}{{(n + 7)}^{2}}$

Step 2.6.2

Multiply $- 1$ by $1$ .

$\frac{n + 7 - n}{{(n + 7)}^{2}}$

Step 2.6.3

Subtract $n$ from $n$ .

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

Step 2.6.4

Add $0$ and $7$ .

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