Calculus Examples

4 - x^{2}

$4 - x^{2}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of

4 - x^{2}

$4 - x^{2}$ with respect to

x

$x$ is

\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]

$\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$ .

\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]

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

Step 1.2

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4

$4$ with respect to

x

$x$ is

0

$0$ .

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

$0 + \frac{d}{d x} [- x^{2}]$

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

$0 + \frac{d}{d x} [- x^{2}]$

Step 2

Evaluate

\frac{d}{d x} [- x^{2}]

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

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Step 2.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- x^{2}

$- x^{2}$ with respect to

x

$x$ is

- \frac{d}{d x} [x^{2}]

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

0 - \frac{d}{d x} [x^{2}]

$0 - \frac{d}{d x} [x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

0 - (2 x)

$0 - (2 x)$

Step 2.3

Multiply

2

$2$ by

- 1

$- 1$ .

0 - 2 x

$0 - 2 x$

0 - 2 x

$0 - 2 x$

Step 3

Subtract

2 x

$2 x$ from

0

$0$ .

- 2 x

$- 2 x$

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$