Question

Using the limit definition, how do you differentiate `f(x) =4/(sqrt(x))`?

Answer 1

`f'(x)=-2/x^(3/2)`

Explanation:

The limit definition of the derivative:

`lim_(hrarr0)(f(x+h)-f(x))/h`

Thus, the derivative can be found through

`f'(x)=lim_(hrarr0)(4/sqrt(x+h)-4/sqrtx)/h`

`=lim_(hrarr0)(4/sqrt(x+h)-4/sqrtx)/h*(sqrtx * sqrt(x+h))/(sqrtx * sqrt(x+h))`

`=lim_(hrarr0)(4sqrtx-4sqrt(x+h))/(hsqrtxsqrt(x+h))`

`=lim_(hrarr0)(4(sqrtx-sqrt(x+h)))/(hsqrtxsqrt(x+h))*(sqrtx+sqrt(x+h))/(sqrtx+sqrt(x+h))`

`=lim_(hrarr0)(4(x-(x+h)))/(hsqrtxsqrt(x+h)(sqrtx+sqrt(x+h)))`

`=lim_(hrarr0)(-4h)/(hsqrtxsqrt(x+h)(sqrtx+sqrt(x+h)))`

`=lim_(hrarr0)(-4)/(sqrtxsqrt(x+h)(sqrtx+sqrt(x+h)))`

Now, plug in `0` for `h`.

`=(-4)/(sqrtxsqrtx(sqrtx+sqrtx))`

`=(-4)/(x(2sqrtx)`

`=color(green)(-2/x^(3/2)`