Calculus Examples

$\frac{x y}{3 x + 8 y}$

Step 1

Since $x$ is constant with respect to $y$ , the derivative of $\frac{x y}{3 x + 8 y}$ with respect to $y$ is $x \frac{d}{d y} [\frac{y}{3 x + 8 y}]$ .

$x \frac{d}{d y} [\frac{y}{3 x + 8 y}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = y$ and $g (y) = 3 x + 8 y$ .

$x \frac{(3 x + 8 y) \frac{d}{d y} [y] - y \frac{d}{d y} [3 x + 8 y]}{{(3 x + 8 y)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$x \frac{(3 x + 8 y) \cdot 1 - y \frac{d}{d y} [3 x + 8 y]}{{(3 x + 8 y)}^{2}}$

Step 3.2

Multiply $3 x + 8 y$ by $1$ .

$x \frac{3 x + 8 y - y \frac{d}{d y} [3 x + 8 y]}{{(3 x + 8 y)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $3 x + 8 y$ with respect to $y$ is $\frac{d}{d y} [3 x] + \frac{d}{d y} [8 y]$ .

$x \frac{3 x + 8 y - y (\frac{d}{d y} [3 x] + \frac{d}{d y} [8 y])}{{(3 x + 8 y)}^{2}}$

Step 3.4

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

$x \frac{3 x + 8 y - y (0 + \frac{d}{d y} [8 y])}{{(3 x + 8 y)}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d y} [8 y]$ .

$x \frac{3 x + 8 y - y \frac{d}{d y} [8 y]}{{(3 x + 8 y)}^{2}}$

Step 3.6

Since $8$ is constant with respect to $y$ , the derivative of $8 y$ with respect to $y$ is $8 \frac{d}{d y} [y]$ .

$x \frac{3 x + 8 y - y (8 \frac{d}{d y} [y])}{{(3 x + 8 y)}^{2}}$

Step 3.7

Multiply $8$ by $- 1$ .

$x \frac{3 x + 8 y - 8 y \frac{d}{d y} [y]}{{(3 x + 8 y)}^{2}}$

Step 3.8

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

$x \frac{3 x + 8 y - 8 y \cdot 1}{{(3 x + 8 y)}^{2}}$

Step 3.9

Simplify terms.

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Step 3.9.1

Multiply $- 8$ by $1$ .

$x \frac{3 x + 8 y - 8 y}{{(3 x + 8 y)}^{2}}$

Step 3.9.2

Subtract $8 y$ from $8 y$ .

$x \frac{3 x + 0}{{(3 x + 8 y)}^{2}}$

Step 3.9.3

Add $3 x$ and $0$ .

$x \frac{3 x}{{(3 x + 8 y)}^{2}}$

Step 3.9.4

Combine $x$ and $\frac{3 x}{{(3 x + 8 y)}^{2}}$ .

$\frac{x (3 x)}{{(3 x + 8 y)}^{2}}$

Step 4

Raise $x$ to the power of $1$ .

$\frac{3 (x^{1} x)}{{(3 x + 8 y)}^{2}}$

Step 5

Raise $x$ to the power of $1$ .

$\frac{3 (x^{1} x^{1})}{{(3 x + 8 y)}^{2}}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 x^{1 + 1}}{{(3 x + 8 y)}^{2}}$

Step 7

Add $1$ and $1$ .

$\frac{3 x^{2}}{{(3 x + 8 y)}^{2}}$