Calculus Examples

$\frac{- x}{4 y}$

Step 1

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 $f (y) = - x$ and $g (y) = 4 y$ .

$\frac{4 y \frac{d}{d y} [- x] - - 1 x \frac{d}{d y} [4 y]}{{(4 y)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{4 y \cdot 0 - - 1 x \frac{d}{d y} [4 y]}{{(4 y)}^{2}}$

Step 2.2

Since $4$ is constant with respect to $y$ , the derivative of $4 y$ with respect to $y$ is $4 \frac{d}{d y} [y]$ .

$\frac{4 y \cdot 0 - - 1 x (4 \frac{d}{d y} [y])}{{(4 y)}^{2}}$

Step 2.3

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

$\frac{4 y \cdot 0 - - 1 x (4 \cdot 1)}{{(4 y)}^{2}}$

Step 2.4

Multiply $4$ by $1$ .

$\frac{4 y \cdot 0 - - 1 x \cdot 4}{{(4 y)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $4 y$ .

$\frac{4 y \cdot 0 - - 1 x \cdot 4}{4^{2} y^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $4 y \cdot 0$ .

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Step 3.2.1.1.1

Multiply $0$ by $4$ .

$\frac{0 y - - 1 x \cdot 4}{4^{2} y^{2}}$

Step 3.2.1.1.2

Multiply $0$ by $y$ .

$\frac{0 - - 1 x \cdot 4}{4^{2} y^{2}}$

Step 3.2.1.2

Multiply $- 1$ by $- 1$ .

$\frac{0 + 1 x \cdot 4}{4^{2} y^{2}}$

Step 3.2.1.3

Multiply $x$ by $1$ .

$\frac{0 + x \cdot 4}{4^{2} y^{2}}$

Step 3.2.1.4

Move $4$ to the left of $x$ .

$\frac{0 + 4 x}{4^{2} y^{2}}$

Step 3.2.2

Add $0$ and $4 x$ .

$\frac{4 x}{4^{2} y^{2}}$

Step 3.3

Combine terms.

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Step 3.3.1

Raise $4$ to the power of $2$ .

$\frac{4 x}{16 y^{2}}$

Step 3.3.2

Cancel the common factor of $4$ and $16$ .

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Step 3.3.2.1

Factor $4$ out of $4 x$ .

$\frac{4 (x)}{16 y^{2}}$

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Factor $4$ out of $16 y^{2}$ .

$\frac{4 (x)}{4 (4 y^{2})}$

Step 3.3.2.2.2

Cancel the common factor.

$\frac{4 x}{4 (4 y^{2})}$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{x}{4 y^{2}}$