Find the Function f(x)=(1/3)^x

$f (x) = {(\frac{1}{3})}^{x}$

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

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

The integral of ${(\frac{1}{3})}^{x}$ with respect to $x$ is $\frac{{(\frac{1}{3})}^{x}}{\ln (\frac{1}{3})}$ .

$\frac{{(\frac{1}{3})}^{x}}{\ln (\frac{1}{3})} + C$

Simplify the answer.

Apply the product rule to $\frac{1}{3}$ .

$\frac{\frac{1^{x}}{3^{x}}}{\ln (\frac{1}{3})} + C$

Simplify.

One to any power is one.

$\frac{\frac{1}{3^{x}}}{\ln (\frac{1}{3})} + C$

Rewrite $\frac{\frac{1}{3^{x}}}{\ln (\frac{1}{3})}$ as a product.

$\frac{1}{3^{x}} \cdot \frac{1}{\ln (\frac{1}{3})} + C$

Multiply $\frac{1}{3^{x}}$ by $\frac{1}{\ln (\frac{1}{3})}$ .

$\frac{1}{3^{x} \ln (\frac{1}{3})} + C$

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

$F (x) = \frac{1}{3^{x} \ln (\frac{1}{3})} + C$