Calculus Examples

$y = x (x - 2)$

Differentiate both sides of the equation.

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

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

$y'$

Differentiate the right side of the equation.

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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 $g (x) = x - 2$ .

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

Differentiate.

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

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

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

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

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

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

Simplify the expression.

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Add $1$ and $0$ .

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

Multiply $x$ by $1$ .

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

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

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

Simplify by adding terms.

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Multiply $x - 2$ by $1$ .

$x + (x - 2)$

Remove unnecessary parentheses.

$x + x - 2$

Add $x$ and $x$ .

$2 x - 2$

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

$y' = 2 x - 2$

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

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

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

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Rewrite the equation as $2 x - 2 = 0$ .

$2 x - 2 = 0$

Add $2$ to both sides of the equation.

$2 x = 2$

Divide each term by $2$ and simplify.

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

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Divide $2$ by $2$ .

$x = 1$

Solve for $y$ .

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Remove the extra parentheses.

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

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

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Multiply $1 - 2$ by $1$ .

$y = 1 - 2$

Subtract $2$ from $1$ .

$y = - 1$

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

$(1, - 1)$

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