Calculus Examples

$y = x^{2}$

Step 1

Write $y = x^{2}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

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

$f' (x) = 2 x$

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

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

$2 \cdot 1$

Step 2.2.3

Multiply $2$ by $1$ .

$f'' (x) = 2$

Step 2.3

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

$2$

Step 3

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

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Step 3.1

Set the second derivative equal to $0$ .

$2 = 0$

Step 3.2

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

No solution

Step 4

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

No Inflection Points