Calculus Examples

$f (x) = - x^{2} + 2 x + 6$

Find the second derivative.

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Find the first derivative.

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

$f' (x) = \frac{d}{d x} (- x^{2}) + \frac{d}{d x} (2 x) + \frac{d}{d x} (6)$

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

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Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$f' (x) = - \frac{d}{d x} x^{2} + \frac{d}{d x} (2 x) + \frac{d}{d x} (6)$

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

$f' (x) = - (2 x) + \frac{d}{d x} (2 x) + \frac{d}{d x} (6)$

Multiply $2$ by $- 1$ .

$f' (x) = - 2 x + \frac{d}{d x} (2 x) + \frac{d}{d x} (6)$

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

$f' (x) = - 2 x + 2 \frac{d}{d x} (x) + \frac{d}{d x} (6)$

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

$f' (x) = - 2 x + 2 \cdot 1 + \frac{d}{d x} (6)$

Multiply $2$ by $1$ .

$f' (x) = - 2 x + 2 + \frac{d}{d x} (6)$

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

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

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

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

Find the second derivative.

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

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

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

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

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

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

Multiply $- 2$ by $1$ .

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

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

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

Add $- 2$ and $0$ .

$f'' (x) = - 2$

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

$- 2$

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

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Rewrite the equation as $0 = - 2$ .

$0 = - 2$

Since $0 \neq - 2$ , there are no solutions.

No solution

No values found that can make the second derivative equal to $0$ .

No Inflection Points

There are no inflection points and the interval to test the concavity on is $(- \infty, \infty)$ .

$(- \infty, \infty)$

The graph is concave down because the second derivative is negative.

The graph is concave down

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