Calculus Examples

$x = \frac{3 r + 2}{2 r + 3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d r} (x) = \frac{d}{d r} (\frac{3 r + 2}{2 r + 3})$

Step 2

The derivative of $x$ with respect to $r$ is $x'$ .

$x'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $f (r) = 3 r + 2$ and $g (r) = 2 r + 3$ .

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

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $3$ by $1$ .

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

Step 3.2.5

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

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

Step 3.2.6

Simplify the expression.

Tap for more steps...

Step 3.2.6.1

Add $3$ and $0$ .

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

Step 3.2.6.2

Move $3$ to the left of $2 r + 3$ .

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

Step 3.2.7

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

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Multiply $2$ by $1$ .

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

Step 3.2.11

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

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

Step 3.2.12

Simplify the expression.

Tap for more steps...

Step 3.2.12.1

Add $2$ and $0$ .

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

Step 3.2.12.2

Multiply $2$ by $- 1$ .

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

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Simplify the numerator.

Tap for more steps...

Step 3.3.3.1

Combine the opposite terms in $3 (2 r) + 3 \cdot 3 - 2 (3 r) - 2 \cdot 2$ .

Tap for more steps...

Step 3.3.3.1.1

Reorder the factors in the terms $3 (2 r)$ and $- 2 (3 r)$ .

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

Step 3.3.3.1.2

Subtract $2 \cdot 3 r$ from $2 \cdot 3 r$ .

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

Step 3.3.3.1.3

Add $3 \cdot 3$ and $0$ .

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

Step 3.3.3.2

Simplify each term.

Tap for more steps...

Step 3.3.3.2.1

Multiply $3$ by $3$ .

$\frac{9 - 2 \cdot 2}{{(2 r + 3)}^{2}}$

Step 3.3.3.2.2

Multiply $- 2$ by $2$ .

$\frac{9 - 4}{{(2 r + 3)}^{2}}$

Step 3.3.3.3

Subtract $4$ from $9$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $x'$ with $\frac{d x}{d r}$ .

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