Unit 2 - Objective 3 - Implicit Differentation




y = 2x² + 6x - 3 is an example where y is an explicit function of x and dy/dx = 4x+6.
On the other hand, x² + y² = 4 has y as an implied function of x. Rather than trying to solve this equation for y and then finding the derivative, we can take the derivative of this equation the way it is. We can do this by remembering that y = u, an implied function of x, and using the chain rule to find the derivative of

Chain Rule

In x² + y² = 4 we first take the derivative of each term.



Then solve for dy/dx (get all the terms which involve dy/dx on one side of the equation then factor it out, and divide through by its coefficient).




Problem 1:

Problem 2:


Problem 3:


This is the slope of the tangent line to the curve at any point. If you pick a point on the curve, you can find the slope of the tangent line to the curve at that point. Say for example, you pick the point (-7,1) on the curve of Example 3. Then the slope of the tangent line would be:



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