Calculus - Implicit differentiation

Question

How would I solve this question to find dy/dx? Would I try to put each x and y variable on separate sides of the equation then product rule to find dy/dx? Any tips would be helpful. Thanks!

y³ + xy = x⁵

Answer 1

Because of the "xy" term in the equation, you're not going to be able to separate the variables to each side of the equation, but that's OK; this is exactly the type of situation in which implicit differentiation works well!

You need to differentiate each term with respect to y and yes, you will need product rule for that middle term. Once you differentiate each term, then you can move all the terms with a dy/dx in them to one side and isolate from there. Hope this helps! Leave me a comment if you need more help than this.

Answer 2

If you could solve for y in terms of x, then find dy/dx you would be doing so explicitly. For some equations/relations it is not possible to solve for y in terms of x. So use implicit differentiation. The idea is to assume that y is a function of x, so that when you take the derivative of y with respect to x you are employing the chain rule.

When you find the derivative of y³, for example, you are envisioning (f(x))³, so the derivative with respect to x is: 3[f(x)]² f'(x), i.e. the chain rule. Frequently we write this as 3y²y'. or 3y²dy/dx. I prefer y' myself.

So,

3y²y' + xy' + y = 5x⁴ (note that we used the product rule for the xy term).

Now get terms containing y' on one side of the equal sign and all other terms on the other side.

y'(3y² + x) = 5x⁴-y (note how the y' was factored out from the two terms containing y')

Finally,

y' = (5x⁴-y) / (3y²+x)