Question

What is the derivative of `sqrt x^2`?

Answer 1

`x/sqrt(x^2)`

Explanation:

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

now differentiate using the chain rule :

`1/2(x^2)^(-1/2).d/dx(x^2)`

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

It is usual to give answer in the same form as question.

Answer 2

Assuming the question is asking `(sqrt(x))^2` or `sqrt((x^2))`, which is then simplified to `x`, the answer is `1`.

Explanation:

`(sqrt(x))^2` can be rewritten as `(x^(1/2))^(2)`. Multiplying the exponents gives `x^1`, which is just `x`.

The definition of `d/dx(x) = 1` is typically memorized, though you can always use the power rule to get the same answer:

`d/dx(x) = d/dx(x^1) = (1)x^(1-1) = (1)(x^0) = (1)(1) = 1`

If the problem were instead asking `d/dx(sqrt(x))`, with the `2` indicating that it's the second root, then the same power rule approach would be taken:

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