Question

How do you find the derivative of `x+sqrt(x)`?

Answer 1

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

Explanation:

Recall the following:

`1`. `color(red)("Sum Rule")`: `[f(x)+g(x)]'=f'(x)+g'(x)`
`2`. `color(blue)("Power Rule")`: `(x^n)'=color(orange)nx^(color(orange)n-1)`

Finding the Derivative
`1`. Start by rewriting `sqrt(x)` as `x^(1/2)`. Recall that `x` is `x^1`.

`(x+sqrt(x))'`

`=(x^1+x^(1/2))'`

`2`. Using the `color(red)("sum rule")`, determine the derivative of each term in the expression. To do that, use the `color(blue)("power rule")`.

`=color(orange)1x^(color(orange)1-1)+color(orange)(1/2)x^(color(orange)(1/2)-1)`

`3`. Simplify.

`=color(orange)1x^0+color(orange)(1/2)x^(-1/2)`

`=color(orange)1(1)+color(orange)(1/2)(1/x)^(1/2)`

`=1+color(orange)(1/2)(1/sqrt(x))`

`color(green)(=1+1/(2sqrt(x)))`

`:.`, the derivative is `1+1/(2sqrt(x))`.