Question

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

Answer 1

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

Explanation:

Chain Rule is not needed here. Just use thee Power Rule.

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

Answer 2

Use algebra and the power rule.

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

Explanation:

When `f(x) = x^n`, the power rule states that the derivative follows the trend `f'(x) = nx^(n-1)`.

Applied to your specific problem...

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

(Use your knowledge of algebra to rewrite the root as an exponent...)

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

(Now just take the derivative by using the power rule...)

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