Calculus Examples

$f (x) = x^{2} - 2 x + 8$

Find the first derivative.

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Find the first derivative.

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Differentiate.

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By the Sum Rule, the derivative of $x^{2} - 2 x + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [8]$ .

$f' (x) = \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (8)$

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

$f' (x) = 2 x + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (8)$

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

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

$f' (x) = 2 x - 2 \frac{d}{d x} x + \frac{d}{d x} (8)$

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

$f' (x) = 2 x - 2 \cdot 1 + \frac{d}{d x} (8)$

Multiply $- 2$ by $1$ .

$f' (x) = 2 x - 2 + \frac{d}{d x} (8)$

Differentiate using the Constant Rule.

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Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f' (x) = 2 x - 2 + 0$

Add $2 x - 2$ and $0$ .

$f' (x) = 2 x - 2$

The first derivative of $f (x)$ with respect to $x$ is $2 x - 2$ .

$2 x - 2$

Set the first derivative equal to $0$ then solve the equation $2 x - 2 = 0$ .

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Set the first derivative equal to $0$ .

$2 x - 2 = 0$

Add $2$ to both sides of the equation.

$2 x = 2$

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

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

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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Divide $2$ by $2$ .

$x = 1$

Find the values where the derivative is undefined.

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The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Evaluate $x^{2} - 2 x + 8$ at each $x$ value where the derivative is $0$ or undefined.

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Evaluate at $x = 1$ .

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Substitute $1$ for $x$ .

${(1)}^{2} - 2 \cdot 1 + 8$

Simplify.

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Simplify each term.

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One to any power is one.

$1 - 2 \cdot 1 + 8$

Multiply $- 2$ by $1$ .

$1 - 2 + 8$

Simplify by adding and subtracting.

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Subtract $2$ from $1$ .

$- 1 + 8$

Add $- 1$ and $8$ .

$7$

List all of the points.

$(1, 7)$