Find the Function f(x)=(1/2)^x

$f (x) = {(\frac{1}{2})}^{x}$

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

The integral of ${(\frac{1}{2})}^{x}$ with respect to $x$ is $\frac{{(\frac{1}{2})}^{x}}{\ln (\frac{1}{2})}$ .

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

Step 3

Simplify the answer.

Apply the product rule to $\frac{1}{2}$ .

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

Simplify.

One to any power is one.

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

Rewrite $\frac{\frac{1}{2^{x}}}{\ln (\frac{1}{2})}$ as a product.

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

Multiply $\frac{1}{2^{x}}$ by $\frac{1}{\ln (\frac{1}{2})}$ .

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

Step 4

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

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