Calculus Examples

$f' (x) = 2 x$

Step 1

The function $f (x)$ can be found by evaluating the indefinite integral of the derivative $f' (x)$ .

$f (x) = \int f' (x) d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x d x$

Step 3

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$2 (\frac{1}{2} x^{2} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $2 (\frac{1}{2} x^{2} + C)$ as $2 (\frac{1}{2}) x^{2} + C$ .

$2 (\frac{1}{2}) x^{2} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} x^{2} + C$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{2}{2} x^{2} + C$

Step 4.2.2.2

Rewrite the expression.

$1 x^{2} + C$

Step 4.2.3

Multiply $x^{2}$ by $1$ .

$x^{2} + C$

Step 5

The function $f$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

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