Calculus Examples

$y = x (x - 2)$

Step 1

Differentiate both sides of the equation.

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

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 using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - 2$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

$x (1 + \frac{d}{d x} [- 2]) + (x - 2) \frac{d}{d x} [x]$

Step 3.2.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$x (1 + 0) + (x - 2) \frac{d}{d x} [x]$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

$x \cdot 1 + (x - 2) \frac{d}{d x} [x]$

Step 3.2.4.2

Multiply $x$ by $1$ .

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

Step 3.2.5

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

$x + (x - 2) \cdot 1$

Step 3.2.6

Simplify by adding terms.

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Step 3.2.6.1

Multiply $x - 2$ by $1$ .

$x + x - 2$

Step 3.2.6.2

Add $x$ and $x$ .

$2 x - 2$

Step 4

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

$y' = 2 x - 2$

Step 5

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

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

Step 6

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 6.1

Add $2$ to both sides of the equation.

$2 x = 2$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $2$ by $2$ .

$x = 1$

Step 7

Solve for $y$ .

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Step 7.1

Remove parentheses.

$y = 1 (1 - 2)$

Step 7.2

Remove parentheses.

$y = (1) ((1) - 2)$

Step 7.3

Simplify $(1) ((1) - 2)$ .

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Step 7.3.1

Multiply $(1) - 2$ by $1$ .

$y = (1) - 2$

Step 7.3.2

Subtract $2$ from $1$ .

$y = - 1$

Step 8

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

$(1, - 1)$

Step 9

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