Calculus Examples

$y = 9 - x^{2}$

Step 1

Reorder $9$ and $- x^{2}$ .

$y = - x^{2} + 9$

Step 2

Set $y$ as a function of $x$ .

$f (x) = - x^{2} + 9$

Step 3

Find the derivative.

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Step 3.1

By the Sum Rule, the derivative of $- x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [9]$

Step 3.2

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

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Step 3.2.1

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

$- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

Step 3.2.2

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

$- (2 x) + \frac{d}{d x} [9]$

Step 3.2.3

Multiply $2$ by $- 1$ .

$- 2 x + \frac{d}{d x} [9]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$- 2 x + 0$

Step 3.3.2

Add $- 2 x$ and $0$ .

$- 2 x$

Step 4

Divide each term in $- 2 x = 0$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 x = 0$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{0}{- 2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{0}{- 2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- 2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 2$ .

$x = 0$

Step 5

Solve the original function $f (x) = - x^{2} + 9$ at $x = 0$ .

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - {(0)}^{2} + 9$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = - 0 + 9$

Step 5.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = 0 + 9$

Step 5.2.2

Add $0$ and $9$ .

$f (0) = 9$

Step 5.2.3

The final answer is $9$ .

$9$

Step 6

The horizontal tangent line on function $f (x) = - x^{2} + 9$ is $y = 9$ .

$y = 9$

Step 7