Question

Find the derivative of `x+sqrtx` using the definition of derivative?

Answer 1

`d/(dx)(x+sqrtx)=1+1/(2sqrtx)`

Explanation:

`(df)/(dx)` is defined as `Lt_(h->0)(f(x+h)-f(x))/h`

Here `f(x)=x+sqrtx` and hence

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

`f(x+h)-f(x)=h+sqrt(x+h)-sqrtx` and hence

`(df)/(dx)=Lt_(h->0)[1+(sqrt(x+h)-sqrtx)/h]`

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

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

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

= `1+1/(2sqrtx)`