7. Implicit Differentiation

Implicit Functions All the methods we have covered so far have had y solved in terms of x, for example y = x3+1 What if instead we have an equation like y4 + xy = x3 – x + 2 In this case, we say y is defined implicitly

Implicit Differentiation Used when an equation cannot easily be solved for y Steps Find dy/dx of both sides of the equation (in other words, the derivative with respect to x for both sides of the equation) Whenever there is a y, you must put a dy/dx along with the derivative Solve for dy/dx

Example 1

What if instead this said Example 2 What if instead this said

Find the tangent line to the ellipse below at (-1,2) Example 3 Find the tangent line to the ellipse below at (-1,2)

Horizontal and Vertical Tangents When the derivative is a ratio of 2 variable expressions Horizontal tangents occur when numerator = 0 Vertical tangents occur when denominator = 0 No tangent exists when both = 0

Example 4 Find the coordinates where the graph of has horizontal and vertical tangents. At what point is the slope ¾?

Example 5 Find the coordinates where the graph of has horizontal and vertical tangents.

Example 6 Find dy/dx if

Finding a 2nd derivative implicitly Take 2nd derivative and plug in y’ Example 7