implicit derivative after multiplying by x

Question

implicit derivative of y/x=10 is y'=y/x. but implicit derivative of y=10x (multiply both sides by x first) is simply y'=10. What is the difference?

Answer 1

To find y' when y/x = 10, we can differentiate implicitly or explicitly.

 

Implicit:   (xy' - y)/(x²) = 0  So, xy' - y = 0

 

                                                    xy' = y

 

                                                      y' = y/x = 10

 

Explicit:  Since y/x = 10, y = 10x   So, y' = 10

 

We get the same result either way.

 

Answer 2

Implicit differentiate means to derivative both sides of the equation using the chain rule.

 

(y'x - y) / x² = 0

 

 

Multiply both sides of the equation by x².

 

y'x - y = 0

 

y'x = y

 

y' = y/x

 

y = 10x  ---->  substitute into derivative

 

y' = 10x / x

 

y' = 10

 

 

 

Now for y=10x:

 

Derive both sides of the equation,

 

y' = 10

 

 

The difference is that you arrange the equations so that in one equation, y is explicitly defined by x.  The other has y defined implicitly with x.