3.7 HW Problems

Table of Contents:

Implicit differentiation (example walkthrough) (video) | Khan Academy

<aside> 💡 Sal Khan, “What I want you to keep in the back of your mind the entire time is that it’s just an application of the Chain Rule.”

</aside>

• Most of the functions we have dealt with so far have been described by an equation of the form $y=f(x)$ that expresses y EXPLICITLY in terms of the variable x.
• A different situation occurs when we encounter equations like:

$$ x^3+y^3-9xy=0 $$

$$ y^2-x=0 $$

$$ x^2+y^2-25=0 $$

• These equations define an implicit relation between the variables x and y.
  • meaning that a value of x makes up one or more values of y

This curve, $x^3+y^3-9xy=0$, is NOT the graph as any one function (as we known via the vertical line test). But this curve can be divided into separate arcs that are graphs of functions of x.

Implicitly Defined Functions

• Examples:
• To calculate the derivatives of other implicitly defined functions, (referring to the Examples) we want to treat y as a differentiable implicit function of x and apply the same-ole same-ole rules we have used in previous lessons to differentiate BOTH sides of the defining equation.

Implicit Differentiation