Calculus Examples

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

Step 1

Factor $2$ out of $8 - 2 x$ .

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Step 1.1

Factor $2$ out of $8$ .

$\frac{d}{d x} [\frac{2 (4) - 2 x}{3 y^{2}}]$

Step 1.2

Factor $2$ out of $- 2 x$ .

$\frac{d}{d x} [\frac{2 (4) + 2 (- x)}{3 y^{2}}]$

Step 1.3

Factor $2$ out of $2 (4) + 2 (- x)$ .

$\frac{d}{d x} [\frac{2 (4 - x)}{3 y^{2}}]$

Step 2

Since $\frac{2}{3 y^{2}}$ is constant with respect to $x$ , the derivative of $\frac{2 (4 - x)}{3 y^{2}}$ with respect to $x$ is $\frac{2}{3 y^{2}} \frac{d}{d x} [4 - x]$ .

$\frac{2}{3 y^{2}} \frac{d}{d x} [4 - x]$

Step 3

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

$\frac{2}{3 y^{2}} (\frac{d}{d x} [4] + \frac{d}{d x} [- x])$

Step 4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 5

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{2}{3 y^{2}} \frac{d}{d x} [- x]$

Step 6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 7

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

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

Step 8

Combine fractions.

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Step 8.1

Multiply $- 1$ by $1$ .

$\frac{2}{3 y^{2}} \cdot - 1$

Step 8.2

Combine $\frac{2}{3 y^{2}}$ and $- 1$ .

$\frac{2 \cdot - 1}{3 y^{2}}$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $2$ by $- 1$ .

$\frac{- 2}{3 y^{2}}$

Step 8.3.2

Move the negative in front of the fraction.

$- \frac{2}{3 y^{2}}$