Question

Find the derivative by the limit process.
f(x) = 5/x^2

Answer 1

\[\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0} \frac{(\frac{5}{(x+h)^2})-\frac{5}{x^2}}{h} = \lim_{h\rightarrow 0} \frac{\frac{5(x^2)}{x^2(x+h)^2}-\frac{5(x+h)^2}{(x+h)^2x^2}}{h}=\]\[\lim_{h\rightarrow 0} \frac{5(x^2)-5(x+h)^2}{x^2(x+h)^2h}\]and just grind out the rest of the algebra by expanding the numerator and denominator of the fraction and simplifying. You'll have a common factor of \(h\) in the numerator and denominator that will cancel out, and the rest you can evaluate at \(h=0\) to get the limit.