Calculus Examples

$q (r) = \frac{3 r}{5 r + 2}$

Step 1

Since $3$ is constant with respect to $r$ , the derivative of $\frac{3 r}{5 r + 2}$ with respect to $r$ is $3 \frac{d}{d r} [\frac{r}{5 r + 2}]$ .

$3 \frac{d}{d r} [\frac{r}{5 r + 2}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d r} [\frac{f (r)}{g (r)}]$ is $\frac{g (r) \frac{d}{d r} [f (r)] - f (r) \frac{d}{d r} [g (r)]}{{g (r)}^{2}}$ where $f (r) = r$ and $g (r) = 5 r + 2$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $5 r + 2$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $5$ by $1$ .

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

Step 3.7

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

$3 \frac{5 r + 2 - r (5 + 0)}{{(5 r + 2)}^{2}}$

Step 3.8

Simplify terms.

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Step 3.8.1

Add $5$ and $0$ .

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

Step 3.8.2

Multiply $5$ by $- 1$ .

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

Step 3.8.3

Subtract $5 r$ from $5 r$ .

$3 \frac{0 + 2}{{(5 r + 2)}^{2}}$

Step 3.8.4

Add $0$ and $2$ .

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

Step 3.8.5

Combine $3$ and $\frac{2}{{(5 r + 2)}^{2}}$ .

$\frac{3 \cdot 2}{{(5 r + 2)}^{2}}$

Step 3.8.6

Multiply $3$ by $2$ .

$\frac{6}{{(5 r + 2)}^{2}}$