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Calculus Examples

$y = 5 x^{2} + 5$

Step 1

Set

$y$ as a function of

$x$ .

$f (x) = 5 x^{2} + 5$

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of

$5 x^{2} + 5$ with respect to

$x$ is

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [5]$

Step 2.2

Evaluate

$\frac{d}{d x} [5 x^{2}]$ .

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Step 2.2.1

Since

$5$ is constant with respect to

$x$ , the derivative of

$5 x^{2}$ with respect to

$x$ is

$5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$

Step 2.2.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

$5 (2 x) + \frac{d}{d x} [5]$

Step 2.2.3

Multiply

$2$ by

$5$ .

$10 x + \frac{d}{d x} [5]$

$10 x + \frac{d}{d x} [5]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since

$5$ is constant with respect to

$x$ , the derivative of

$5$ with respect to

$x$ is

$0$ .

$10 x + 0$

Step 2.3.2

Add

$10 x$ and

$0$ .

$10 x$

$10 x$

$10 x$

Step 3

Divide each term in

$10 x = 0$ by

$10$ and simplify.

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Step 3.1

Divide each term in

$10 x = 0$ by

$10$ .

$\frac{10 x}{10} = \frac{0}{10}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

$10$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{0}{10}$

Step 3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{0}{10}$

$x = \frac{0}{10}$

$x = \frac{0}{10}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide

$0$ by

$10$ .

$x = 0$

$x = 0$

$x = 0$

Step 4

Solve the original function

$f (x) = 5 x^{2} + 5$ at

$x = 0$ .

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Step 4.1

Replace the variable

$x$ with

$0$ in the expression.

$f (0) = 5 {(0)}^{2} + 5$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising

$0$ to any positive power yields

$0$ .

$f (0) = 5 \cdot 0 + 5$

Step 4.2.1.2

Multiply

$5$ by

$0$ .

$f (0) = 0 + 5$

$f (0) = 0 + 5$

Step 4.2.2

Add

$0$ and

$5$ .

$f (0) = 5$

Step 4.2.3

The final answer is

$5$ .

$5$

$5$

$5$

Step 5

The horizontal tangent line on function

$f (x) = 5 x^{2} + 5$ is

$y = 5$ .

$y = 5$

Step 6

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