Calculus Examples

$f (x) x^{2} - 1$

Step 1

By the Sum Rule, the derivative of $f (x) x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [f (x) x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 2

Evaluate $\frac{d}{d x} [f (x) x^{2}]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Move $2$ to the left of $f (x)$ .

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

Step 3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 f (x) x + 0$

Step 4

Simplify.

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Step 4.1

Add $2 f (x) x$ and $0$ .

$2 f (x) x$

Step 4.2

Reorder the factors of $2 f (x) x$ .

$2 x f (x)$