Question

Question #48bee

Answer 1

`1/(2x)`

Explanation:

To take the derivative of any natural log we take the argument and raise it to the negative first, in this case `1/(sqrt(x))`. However because the argument is not just x we must apply the chain rule to it and multiply the expression we just found by the derivative of the argument alone.

The derivative of the argument `sqrt(x)` can be found using the chain rule and it is `1/(2*sqrt(x))`. Now we multiply both expression together to get: `1/(2*sqrt(x)*sqrt(x))`.

Multiplying two square roots with the same argument however just equals the argument so we can simplify this to the final solution `1/(2x)`. Hope this helped!

Answer 2

`dy/dx = 1/(2x)`

Explanation:

We have:

`y = lnsqrt(x)`

Which we can write as:

`y = lnx^(1/2)`
`\ \ = 1/2lnx` (using the rule of logs)

Differentiating wrt `x` and using `d/dxlnx=1/x`; then:

`dy/dx = 1/2 1/x = 1/(2x)`