What is the Value of dy/dx When y=1 in Implicit Differentiation?

[thereddevils]

Homework Statement

If $$y= \ln \sqrt{xy}$$ , find the value of dy/dx when y=1

Homework Equations

The Attempt at a Solution

$$\frac{dy}{dx} = \frac{1}{\sqrt{xy}} \cdot \frac{1}{2\sqrt{xy}}\cdot (x\frac{dy}{dx}+y)$$

$$\frac{dy}{dx}=\frac{1}{x}$$ , when y=1

 

[CompuChip]

Looks correct so far.
Did you try plugging in y = 1 and simplifying?

 

[HallsofIvy]

You answer should be a number, not a function of x. Since y= ln(xy), when y= 1, 1= ln(x). What is x?

 

[thereddevils]

You answer should be a number, not a function of x. Since y= ln(xy), when y= 1, 1= ln(x). What is x?

this kept me wondering again , if i differentiate first , then plug y=1 in , i get dy/dx=1/x

and now , if i plug y=1 in first , and differentiate later ,

1= ln sqrt(x)

dy/dx=1/2x

Are they supposed to end up with the same result ?

 

[rl.bhat]

In all such problems, you have to differentiate first, then substitute the values.