Calculus Examples

$x (x - 2)$

Step 1

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

$x \frac{d}{d x} [x - 2] + (x - 2) \frac{d}{d x} [x]$

Step 2

Differentiate.

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Step 2.1

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

$x (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [x]$

Step 2.2

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

$x (1 + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [x]$

Step 2.3

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

$x (1 + 0) + (x - 2) \frac{d}{d x} [x]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

Multiply $x$ by $1$ .

$x + (x - 2) \frac{d}{d x} [x]$

Step 2.5

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

$x + (x - 2) \cdot 1$

Step 2.6

Simplify by adding terms.

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Step 2.6.1

Multiply $x - 2$ by $1$ .

$x + x - 2$

Step 2.6.2

Add $x$ and $x$ .

$2 x - 2$