Whenever I look at the solution for the derivative of an implicit function, I see that the product rule is used for terms with two different variables. For example, for the equation $e^{xy^2}$=$x-y$ you have to solve for the derivative of $xy^2$ when taking the derivative of $e^{xy^2}$ and at that step you use the product rule.
I'm confused because I thought that I could just treat $y^2$ as a constant in the term ${xy^2}$ and thus get $d(xy^2)/dx=y^2$. Why can't I do that?
[Math] Implicit differentiation product rule
calculusimplicit-differentiation
Best Answer
What you have to understand is that $y$ is a
function. You can think of it as $y(x)$. Thus, when you have $xy^2$, $y^2$ is actually afunction, which is $[y(x)]^2$. The derivative of $xy^2$ would be:$$(1)(y^2) + (x)(2)(y)(y'(x))$$ $$y^2 + 2xyy'(x)$$
It is
nota constant. Sure, if you have something like $2y$, then the derivative is $2y'(x)$. Notice how if it was $2x$, then its just $2$. But since $y$ is afunction, you must treat is as such.Usually, $y'(x)$, is just abbreviated as $y'$.
Now that you know that, let's differentiate $e^{xy^2}$.
Thats $e^{xy^2}$ * the derivative of ${xy^2}$, which we calculated above. Thus, we have:
$$e^{xy^2} * [y^2 + 2xyy']$$
Differentiating $x - y$ is easy. It's just $1 - y'$.
In the end you have:
$$e^{xy^2} * [y^2 + 2xyy'] = 1 - y'$$