Calculus Examples

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

Step 1

Find the first derivative.

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Differentiate.

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

$f' (x) = \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2 x)$

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

$f' (x) = 2 x + \frac{d}{d x} (- 2 x)$

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

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

$f' (x) = 2 x - 2 \frac{d}{d x} x$

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

$f' (x) = 2 x - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$f' (x) = 2 x - 2$

Step 2

Find the second derivative.

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

$f'' (x) = \frac{d}{d x} (2 x) + \frac{d}{d x} (- 2)$

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

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

$f'' (x) = 2 \frac{d}{d x} (x) + \frac{d}{d x} (- 2)$

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

$f'' (x) = 2 \cdot 1 + \frac{d}{d x} (- 2)$

Multiply $2$ by $1$ .

$f'' (x) = 2 + \frac{d}{d x} (- 2)$

Differentiate using the Constant Rule.

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

$f'' (x) = 2 + 0$

Add $2$ and $0$ .

$f'' (x) = 2$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $2$ .

$2$