Linear Algebra Examples

$f (x) = x^{2} - 2 x - 3$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{2} - 2 x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3]$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 2.2

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

$2 x - 2 \cdot 1 + \frac{d}{d x} [- 3]$

Step 2.3

Multiply $- 2$ by $1$ .

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

Step 3

Differentiate using the Constant Rule.

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Step 3.1

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

$2 x - 2 + 0$

Step 3.2

Add $2 x - 2$ and $0$ .

$2 x - 2$