Answer
$$A^{\prime}(x) = x$$
Work Step by Step
The area of the triangle equals half the base multiplied by the height, and the height equals $x$ while the half base equals $x$, so the area will be
$$A = \dfrac{1}{2} (x)(x) = \dfrac{x^2}{2}$$
If we differentiate the area, we will get the function of $x$ as requested:
$$A^{\prime}(x)=\frac{d}{d x} \left(\dfrac{x^{2}}{ 2}\right) =x=f(x)$$