Evaluating limits with several notations $$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$
It is known that $f'(1) = -2$
Given this info, I'm left with many questions. I'm going to assume that I'll want to substitute for something. I'll let $g(x) = e^{5x} - x^2$. But how do I incorporate the fact that $f'(1) = -2$? Should $g(1) = f'(1)$? 
Also, is the differentiation of g(x) as follows?:
$$f'(e^{5x}-x^2) = 5e^{5x} - 2x $$ 
I figured this is wrong as it's inside the function notation.  

You could also use compose Taylor series to get  $$f(e^{5x}-x^2)=f(1)+5 x f'(1)+x^2 \left(\frac{25 f''(1)}{2}+\frac{23
   f'(1)}{2}\right)+O\left(x^3\right)$$ which makes $$\frac{f(e^{5x} - x^2) - f(1)}{x}=5 f'(1)+x \left(\frac{25 f''(1)}{2}+\frac{23 f'(1)}{2}\right)+O\left(x^2\right)$$
