1. You can do it!!! 2.5 Implicit Differentiation

2. How would you find the derivative in the equation x 2 – 2y 3 + 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. 3x 2 2y 3 x + 3y xy 2 6x 6y 2 y’ 1 + 3y’ Product rule x(2y)y’ + y 2 (1)= 2xyy’ + y 2

4. Find dy/dx given that y 3 + y 2 – 5y – x 2 = -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 x 2 + 4y 2 = 4 at the point. -2 -1 1 2

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 x 2 + y 2 = 25, find y” Now replace y’ with Multiply top and bottom by y What can we substitute in for x 2 + y 2 ?