Calculus Examples

$f' (x) = - 2 x \sqrt{8 - x^{2}}$

Step 1

The function $f (x)$ can be found by evaluating the indefinite integral of the derivative $f' (x)$ .

$f (x) = \int f' (x) d x$

Step 2

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$- 2 \int x \sqrt{8 - x^{2}} d x$

Step 3

Let $u = 8 - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 8 - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $8 - x^{2}$ .

$\frac{d}{d x} [8 - x^{2}]$

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of $8 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}]$

Step 3.1.2.2

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

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

Step 3.1.3

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

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Step 3.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 3.1.3.2

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

$0 - (2 x)$

Step 3.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 3.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 3.2

Rewrite the problem using $u$ and $d u$ .

$- 2 \int \sqrt{u} \frac{1}{- 2} d u$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

$- 2 \int \sqrt{u} (- \frac{1}{2}) d u$

Step 4.2

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

$- 2 \int - \frac{\sqrt{u}}{2} d u$

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- 2 (- \int \frac{\sqrt{u}}{2} d u)$

Step 6

Multiply $- 1$ by $- 2$ .

$2 \int \frac{\sqrt{u}}{2} d u$

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$2 (\frac{1}{2} \int \sqrt{u} d u)$

Step 8

Simplify the expression.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{2}$ and $2$ .

$\frac{2}{2} \int \sqrt{u} d u$

Step 8.1.2

Cancel the common factor of $2$ .

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Step 8.1.2.1

Cancel the common factor.

$\frac{2}{2} \int \sqrt{u} d u$

Step 8.1.2.2

Rewrite the expression.

$1 \int \sqrt{u} d u$

Step 8.1.3

Multiply $\int \sqrt{u} d u$ by $1$ .

$\int \sqrt{u} d u$

Step 8.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int u^{\frac{1}{2}} d u$

Step 9

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{2}{3} u^{\frac{3}{2}} + C$

Step 10

Replace all occurrences of $u$ with $8 - x^{2}$ .

$\frac{2}{3} {(8 - x^{2})}^{\frac{3}{2}} + C$

Step 11

The function $f$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f (x) = \frac{2}{3} \cdot {(8 - x^{2})}^{\frac{3}{2}} + C$