1. Calculus 3.10 Implicit Differentiation
You can do it!!!

2. How would you find the derivative in the equation x2 – 2y3 + 4y = 2
where it is very difficult to express y as a function
of x?
To do this, we use a procedure called implicit
differentiation.
This means that when we differentiate terms
involving x alone, we can differentiate as usual.
But when we differentiate terms involving y, we
must apply the Chain Rule.
Watch the examples very carefully!!!

3. Differentiate the following with respect to x.
2y3
x + 3y
xy2 6x
6y2 y’
1 + 3y’
x(2y)y’ + y2(1)
= 2xyy’ + y2
Product rule

4. Find dy/dx given that y3 + y2 – 5y – x2 = -4
Isolate dy/dx’s
Factor out dy/dx

5. What are the slopes at the following points?
(2,0)
(1,-3)
x = 0
(1,1)
undefined

6. Determine the slope of the tangent line to the
graph of x2 + 4y2 = 4 at the point

7. Differentiate sin y = x
Differentiate x sin y = y cos x
Product Rule
x cos y (y’) + sin y (1) = y (-sin x) + cos x (1)(y’)
x cos y (y’) - cos x (y’) = -sin y - y sin x
y’(x cos y - cos x) = -sin y - y sin x

8. Given x2 + y2 = 25, find y”
Now replace y’ with
Multiply top and bottom by y
What can we substitute in for x2 + y2?