Calculus Examples

$f (x) = x^{2} + 2 x + 1$

Find the derivative.

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By the Sum Rule, the derivative of $x^{2} + 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$

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

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

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

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

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

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

Multiply $2$ by $1$ .

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

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 2 + 0$

Add $2 x + 2$ and $0$ .

$2 x + 2$

Set the derivative equal to $0$ then solve for $x$ .

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Subtract $2$ from 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 $2$ .

$x = - 1$

After finding the point that makes the derivative $f' (x) = 2 x + 2$ equal to $0$ or undefined, the interval to check where $f (x) = x^{2} + 2 x + 1$ is increasing and where it is decreasing is $(- \infty, - 1) \cup (- 1, \infty)$ .

$(- \infty, - 1) \cup (- 1, \infty)$

Substitute a value from the interval $(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $- 2$ in the expression.

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

Simplify the result.

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Multiply $2$ by $- 2$ .

$f' (- 2) = - 4 + 2$

Add $- 4$ and $2$ .

$f' (- 2) = - 2$

The final answer is $- 2$ .

$- 2$

At $x = - 2$ the derivative is $- 2$ . Since this is negative, the function is decreasing on $(- \infty, - 1)$ .

Decreasing on $(- \infty, - 1)$ since $f' (x) < 0$

Substitute a value from the interval $(- 1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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

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

Simplify the result.

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Multiply $2$ by $0$ .

$f' (0) = 0 + 2$

Add $0$ and $2$ .

$f' (0) = 2$

The final answer is $2$ .

$2$

At $x = 0$ the derivative is $2$ . Since this is positive, the function is increasing on $(- 1, \infty)$ .

Increasing on $(- 1, \infty)$ since $f' (x) > 0$

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- 1, \infty)$

Decreasing on: $(- \infty, - 1)$

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