I have the equation below.
If I work backwards and integrate the second line w.r.t. t and then evaluate at t = x, I can get the first line.
However, how do I go from the first expression to the second? I don't think it's differentiating w.r.t. t, since that would result in 0.
ETA: Here's how I got the first line from the second line:
$$\frac{2}{x^3}\int_{0}^{x}(f(t) - f(x) - f'(x)(t-x))dt$$ $$=\frac{2}{x^3}\int_{0}^{x}f(t)dt - \frac{2}{x^3}\int_{0}^{x}f(x) + tf'(x) - xf'(x) dt$$ $$=\frac{2}{x^3}\int_{0}^{x}f(t)dt - \frac{2}{x^3}[tf(x) + \frac{t^2}{2}f'(x) - xtf'(x)]^x_0$$ $$=\frac{2}{x^3}\int_{0}^{x}f(t)dt - \frac{2}{x^3}[xf(x) + \frac{x^2}{2}f'(x) - x^2f'(x)]$$ $$=\frac{2}{x^3}\int_{0}^{x}f(t)dt - \frac{2}{x^3}[xf(x) - \frac{x^2}{2}f'(x)]$$ $$=\frac{2}{x^3}\int_{0}^{x}f(t)dt - \frac{2f(x)}{x^2} + \frac{f'(x)}{x}$$