how to differentiate this function by using the definition of derivation

Question

how to differentiate this function $f(x)=\dfrac1{(x-2)^2}$ by using the definition of derivation

can i get it without having to separate $(x+h-2)^2$ ?

Answer 1

Hint:

You can just use a substitution, such as $u = (x-2)$. Of course, you may also treat $(x-2)$ as a single variable, but substitution might make things a bit easier. The expansion itself isn’t anything scary really, as it all simplifies nicely.

$$f’(x) = \lim_{h \to 0} \frac{\frac{1}{(u+h)^2}-\frac{1}{u^2}}{h}$$

$$f’(x) = \lim_{h \to 0} \frac{\frac{u^2-(u+h)^2}{u^2(u+h)^2}}{h}$$

$$f’(x) = \lim_{h \to 0} \frac{\frac{u^2-u^2-2uh-h^2}{u^2(u+h)^2}}{h}$$

$$f’(x) = \lim_{h \to 0} \frac{\frac{-2uh-h^2}{u^2(u+h)^2}}{h} = f’(x) = \lim_{h \to 0} \frac{\frac{-\color{blue}{h}(2u+h)}{u^2(u+h)^2}}{\color{blue}{h}}$$

Can you take it on from here?