You can do it!!! 2.5 Implicit Differentiation

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!!!

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

Find dy/dx given that y 3 + y 2 – 5y – x 2 = -4 Isolate dy/dx’s Factor out dy/dx

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

Determine the slope of the tangent line to the graph of x 2 + 4y 2 = 4 at the point

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

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 ?