Question

Implicit Differentiation?

a. In your own words, state the guidelines for implicit differentiation.

b. Think about your explanation and use it to solve the following problem (with regards to your explanation):

`\(xy)^2+x^3y^2=3y`

Answer 1

The derivative is:

`(dy)/(dx)=(2xy^2+3x^2y^2)/(3-2x^2y-2x^3y)`

Explanation:

a) Implicit Differentiation is used for an implicitly defined function (or relation). An implicitly defined function/relation is one that does not express `y` as a function of `x`. In other words, it has not been "solved for `y`." Often it may be difficult or impossible to solve it for `y`.

By contrast, explicit differentiation is used for an explicitly defined function or relation, that is one that has been "solved for `y`." For example, `y=+-sqrt(x)` (which is a relation but not a function) can be differentiated explicitly because it expresses `y` in terms of `x`.

To use implicit differentiation, you find the derivative of the expression in terms of `x` using all of the usual rules (product, quotient, etc.), bearing in mind that you will need to use the chain rule for every term that contains a `y` instead of an `x`, because you are finding the derivative in terms of `x`.

b) For the given example, I would start with a bit of algebra to simplify the job:

`(xy)^2+x^3y^2=3y`

`x^2y^2+x^3y^2=3y`

We differentiate in terms of `x`, applying the product rule to the first two terms and the chain rule to every term containing a `y`:

`d/(dx)(x^2y^2+x^3y^2)=d/(dx)(3y)`

`2xy^2+x^2*2y(dy)/(dx)+3x^2y^2+x^3*2y(dy)/(dx)=3(dy)/(dx)`

We now collect all of the terms containing `dy/dx` on one side of the equation and the other terms on the other side.

`2xy^2+3x^2y^2=3(dy)/(dx)-x^2*2y(dy)/(dx)-x^3*2y(dy)/(dx)`

Now we factor out `(dy)/(dx)` and solve.

`2xy^2+3x^2y^2=(dy)/(dx)(3-x^2*2y-x^3*2y)`

`(2xy^2+3x^2y^2)/(3-2x^2y-2x^3y)=(dy)/(dx)`