I can't find an example in my book so I am not sure how I am suppose to do this.
I am trying to find the derivative of $y$ for $y+x\cos(y) = x^2y$
I can't find an example in my book so I am not sure how I am suppose to do this.
I am trying to find the derivative of $y$ for $y+x\cos(y) = x^2y$
When you do implicit differentiation problems, there are three important things to keep in mind.
Keeping these things in mind, I get $ 1 \cdot y^\prime + (\cos(y) - x\sin(y)y^\prime) = (2xy + x^2\cdot 1 \cdot y^\prime), $ where I've included parentheses to show where product rule is taking place.
Now, the whole point of this business was to get $y^\prime$ by itself. So, move everything having to do with $y^\prime$ to one side of the equation and all other terms to the other. $ y^\prime - x\sin(y)y^\prime - x^2y^\prime = 2xy - \cos(y). $ Factoring out the $y^\prime$ gives $ y^\prime(1 - x\sin(y) - x^2) = 2xy - \cos(y). $ Finally, dividing to isolate $y^\prime$ leaves us with $ y^\prime = \frac{2xy - \cos(y)}{1 - x\sin(y) - x^2}. $