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Now it's time to throw a monkey wrench into the works, curve ball style. What happens when we mix two variables together, on both sides of the equation? Why calculus? Why?! We're entering the Twilight Zone; we're traveling to another dimension. Once the room stops spinning, we'll move on. Good now? Good. We can handle this.
A function is a special kind of relation where each input has exactly one output. Often we have a rule or a formula that tells us how to reach the input to the output, such as
y = x + 5.
In such cases we say the function is defined explicitly. The dependent variable appears all by itself on one side of the equation, and the other side gives a recipe for making the dependent variable from the independent variable.
We say a function is defined implicitly if we have an equation in which the dependent variable doesn't have a side of the equation all to itself. The equation
5x2 – 4y = 7
is an implicit definition of y, since it doesn't actually say how to go from x to y. If we solve this equation for y we find the explicit definition
We can say that y is "defined implicitly" even if y isn't actually a function. For example, the equation
x2 + y2 = 1
describes a circle of radius 1. A circle isn't a function, since it fails the vertical line test. However, we would still say that this equation implicitly defines y.
For a total headache, we might even say "y is an implicitly defined function" even though y isn't a function. We probably won't worry about this distinction in class, but we wanted to point it out for the sake of completeness.
We've taken lots of derivatives of explicit functions. In fact, so far we've only taken derivatives of explicit functions. We've had an equation for y (or f, or whatever) and we've used our collection of rules to find f '.
We can also take derivatives of implicit functions. If we have an equation relating x and y, we can take derivatives first and solve for y' later.
We can use either Leibniz or Lagrange notation.
It will be helpful to have a variation on Lagrange notation. The expression (...) means, "take the derivative, with respect to x, of whatever is in the parentheses." It essentially means the same thing as (...)', but with the added precise statement that yes, we are taking the derivative with respect to x and not with respect to anything else.
Remember, means the same thing as a prime.
(f + g)' = f ' + g '
We can also say
Some things to remember for implicit differentiation:
With these things in mind, we're ready to get cracking.
Not a fan of Leibniz notation? We can do implicit differentiation with Lagrange notation just as well.
Things to remember for implicit differentiation with Lagrange notation: