the sum of the two shorter sides of a right-angled triangle is 18cm. calculate the least possible length of the hypotenuse.
ok so obviously i get sqrt(x^2 + (18-x)^2) but how do i differentiate the whole expression from here?
Hm... maybe going into more details would be better...
It's like differentiating $\displaystyle (f(x))^n$ where $\displaystyle n \neq 1$
The derivative is:
$\displaystyle n (f(x))^{n-1} \times f'(x)$
So, using your problem;
$\displaystyle f(x) = \sqrt{x^2 + (18-x)^2} = (2x^2 -36x + 324)^{\frac{1}{2}}$
$\displaystyle f'(x) = \frac{1}{2} (2x^2 -36x + 324)^{-\frac{1}{2}} \times (4x - 36)$
Which you then simplify to:
$\displaystyle f'(x) = \frac{2x - 18}{\sqrt{2x^2 -36x + 324}}$