Algebra Examples

$\frac{x}{y}$

Step 1

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

$\frac{y \frac{d}{d x} [x] - x \frac{d}{d x} [y]}{y^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

$\frac{y \cdot 1 - x \frac{d}{d x} [y]}{y^{2}}$

Step 2.2

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

$\frac{y \cdot 1 - x \cdot 0}{y^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Multiply $y$ by $1$ .

$\frac{y - x \cdot 0}{y^{2}}$

Step 3.1.1.2

Multiply $- x \cdot 0$ .

Tap for more steps...

Step 3.1.1.2.1

Multiply $0$ by $- 1$ .

$\frac{y + 0 x}{y^{2}}$

Step 3.1.1.2.2

Multiply $0$ by $x$ .

$\frac{y + 0}{y^{2}}$

Step 3.1.2

Add $y$ and $0$ .

$\frac{y}{y^{2}}$

Step 3.2

Cancel the common factor of $y$ and $y^{2}$ .

Tap for more steps...

Step 3.2.1

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

$\frac{y^{1}}{y^{2}}$

Step 3.2.2

Factor $y$ out of $y^{1}$ .

$\frac{y \cdot 1}{y^{2}}$

Step 3.2.3

Cancel the common factors.

Tap for more steps...

Step 3.2.3.1

Factor $y$ out of $y^{2}$ .

$\frac{y \cdot 1}{y \cdot y}$

Step 3.2.3.2

Cancel the common factor.

$\frac{y \cdot 1}{y \cdot y}$

Step 3.2.3.3

Rewrite the expression.

$\frac{1}{y}$