Question #48bee

2 Answers
Mar 22, 2017

#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!

Mar 22, 2017

# 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)#