Implicit Differentiation

I.  Quiz

II.  Homework

III.  Implicit and Explicit functions

An explicit function is an function expressed as y = f(x) such as

y = sinx

y is defined implicitly if both x and y occur on the same side of the equation such as

x² + y² = 4

we can think of y as function of x and write:

x² + y(x)² = 4

IV.  Implicit differentiation

To find dy/dx, we proceed as follows:

1)  Take d/dx of both sides of the equation remembering to multiply by y' each time you see a y term.

d/dx(x² + y²⁾ = d/dx 4

2x + 2yy' = 0

2)  Solve for y'

2yy' = -2x

y' = -2x/2y

y' = -x/y

Example:  Find y' at (2,2) if xy + x/y = 5

Solution:  (1)  (xy)' + (x/y)' = (5)'

Using the product rule and the quotient rule we have

xy' + y + [y - xy']/y² = 0

Now plugging in x = 1 and y = 1,

2y' + 2 + (2 - 2y')/4 = 0 

8y' + 8 + 2 - 2y' = 0

6y' = -10

y' = -5/3

Exercises:

A)  Let    3x² - y³  = 4x cosx + y²

Find dy/dx

B)  Find dy/dx at (-1,1) if

(x + y)³ = x³ + y³  

C) Find dy/dx if

x² + 3xy + y² = 1

D)  Find y'' if

x² - y² = 4