Question

Find the derivative of `f(x)=3sqrtx` using limit definition of derivative?

Answer 1

`d/(dx) 3sqrtx=3/(2sqrtx)`

Explanation:

For a function `f(x)`, limit definition of derivative is

`(df)/(dx)=Lt _(h->0) (f(x+h)-f(x))/h`

Here `f(x)=3sqrtx`

and hence `f(x+h)=3sqrt(x+h)` and hence

`(df)/(dx)=Lt _(h->0) (3sqrt(x+h)-3sqrtx)/h`

= `3Lt_(h->0)((sqrt(x+h)-sqrtx)(sqrt(x+h)+sqrtx))/(h(sqrt(x+h)+sqrtx))`

= `3Lt_(h->0)(x+h-x)/(h(sqrt(x+h)+sqrtx))`

= `3Lt_(h->0)h/(h(sqrt(x+h)+sqrtx))`

= `3Lt_(h->0)1/(sqrt(x+h)+sqrtx)`

= `3xx1/(2sqrtx)`

= `3/(2sqrtx)`