Question

How do you find the derivative of `f(x)= x+ sqrtx`?

Answer 1

`1+(1)/(2sqrt(x))`

Explanation:

We note that derivative of sums and differences can be split apart individually. Meaning,

`d/dx x` + `d/dx sqrt(x)`

where `d/dx x = 1` from the general power rule which states,

`d/dx x^n = nx^(n-1)`

Continuing on this basis,

`sqrt(x) = x^(1/2)`

Since we follow the same rules we first bring down the `1/2` and subtract `1` from `1/2` leaving,

`d/dx sqrt(x) = 1/2 x^(-1/2)`

We further note that we can remove the negative sign of a number by moving it to the opposite of its current location of the form `n/d` where we now move `x^(-1/2)` to the denominator resulting in `(1)/(2sqrt(x))`

Now we simply add the two derivatives together equaling,

`f'(x) = 1+(1)/(2sqrt(x))`

Answer 2

`f'(x) = 1 + 1/2x^(-1/2) = 1 + 1/(2sqrt(x))`

Explanation:

Note the sum rule for derivatives and the power rule:

`d/dx (f(x) + g(x)) = f'(x) + g'(x)`

As such, `d/dx f(x)`, treating `x` and `sqrt(x)` as their own functions in a sense, is...

`d/dx (x + sqrt(x))`

`d/dx x + d/dx sqrt(x)`

Using the power rule for the first term...

`1 + d/dx x^(1/2)`

Using the power rule for the second...

`1 + (1/2)x^((1/2)-1)`

`1 + (1/2)x^(-1/2)`

Which can be rewritten as:

`1 + 1/(2sqrt(x))`