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Dive into the fascinating world of implicit differentiation, a vital technique for finding derivatives when expressing y as a function of x is challenging. Learn how to differentiate complex equations like x² - 2y³ + 4y = 2, applying the Chain Rule thoughtfully when dealing with y. We will cover essential examples, including how to find dy/dx and the slopes of tangent lines at various points. Develop your understanding of the Product Rule and explore deeper concepts like finding second derivatives. Enhance your calculus skills with this comprehensive guide!
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2.5 Implicit Differentiation You can do it!!!
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!!!
Differentiate the following with respect to x. 3x2 2y3 x + 3y xy2 6x 6y2 y’ 1 + 3y’ x(2y)y’ + y2(1) = 2xyy’ + y2 Product rule
Find dy/dx given that y3 + y2 – 5y – x2 = -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 x2 + 4y2 = 4 at the point . -2 -1 1 2
Differentiate sin y = x Product Rule Differentiate x sin y = y cos x 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 x2 + y2 = 25, find y” Now replace y’ with Multiply top and bottom by y What can we substitute in for x2 + y2?
