Calculus Examples

$y = 6 - x^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (6 - x^{2})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $6 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [- x^{2}]$

Step 3.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

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

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

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

Step 3.2.2

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

$0 - (2 x)$

Step 3.2.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 3.3

Subtract $2 x$ from $0$ .

$- 2 x$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - 2 x$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - 2 x$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $- 2$ .

$x = 0$

Step 7

Solve for $y$ .

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Step 7.1

Remove parentheses.

$y = 6 - 0^{2}$

Step 7.2

Simplify $6 - 0^{2}$ .

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

$y = 6 - 0$

Step 7.2.1.2

Multiply $- 1$ by $0$ .

$y = 6 + 0$

Step 7.2.2

Add $6$ and $0$ .

$y = 6$

Step 8

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 6)$

Step 9