Use implicit differentiation to find dy/dx

[domyy]

Homework Statement

Use implicit differentiation to find dy/dx.

Homework Equations

xe^y - 10x + 3y = 0

The Attempt at a Solution

= [xe^y + e^y(y)'] - (10x)' + (3y)' = 0

= xe^y + e^y(y') - 10 + 3(y') = 0

= y' (e^y + 3) = 10 - xe^y

= y' = 10 - xe^y/ e^y + 3y

However, my book says the answer: 10 - e^y/ xe^y + 3y.

What am I doing wrong?

Thanks!

 

[Mark44]

Homework Statement

Use implicit differentiation to find dy/dx.

Homework Equations

xe^y - 10x + 3y = 0

The Attempt at a Solution

The first thing wrong is starting the line below with =. By putting that = at the start of each line, you are more likely to lose track of the two sides of the equation you started with.

= [xe^y + e^y(y)'] - (10x)' + (3y)' = 0

The more serious mistake is above. You didn't take the derivative of xe^y correctly. d/dx(xe^y) = e^y + xe^y*y'. Do you see why?

= xe^y + e^y(y') - 10 + 3(y') = 0

= y' (e^y + 3) = 10 - xe^y

= y' = 10 - xe^y/ e^y + 3y

However, my book says the answer: 10 - e^y/ xe^y + 3y.

No, I believe their answer is (10 - e^y)/(xe^y + 3). Note the parentheses, and the 3 instead of the 3y that you had.

What am I doing wrong?

Thanks!

 

[domyy]

About taking the derivative wrongly, my question is: How does that differ from y= xe^x - e^x?

Because I was trying to follow the same thinking when solving the problem in question.

This is how I solved it ( and I got it right):

= xe^x + e^x (x)' - e^x(x)'
= xe^x + e^x - e^x
= e^x ( x -1 + 1)
= xe^x

Thanks!

 

[Mark44]

About taking the derivative wrongly, my question is: How does that differ from y= xe^x - e^x?

Because I was trying to follow the same thinking when solving the problem in question.

This is how I solved it ( and I got it right):

Don't start your work with =.

= xe^x + e^x (x)' - e^x(x)'
= xe^x + e^x - e^x
= e^x ( x -1 + 1)
= xe^x

Thanks!

In this problem,
d/dx(xe^x) = x *d/dx(e^x) + d/dx(x) * e^x = xe^x + 1*e^x = e^x(x + 1)

The previous problem is different because you need to use the chain rule.
d/dx(xe^y) = x * d/dx(e^y) + d/dx(x) * e^y
= x * d/dy(e^y) * dy/dx + d/dx(x) * e^y = xe^y*y' + 1*e^y = e^y(xy' + 1)

 

[domyy]

Ok, I am going to try to solve some other problems to practice it.