Implicit Differentiation

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Presentation transcript:

Implicit Differentiation

Introduction Consider an equation involving both x and y: This equation implicitly defines a function in x It could be defined explicitly

Differentiate Differentiate both sides of the equation For we get each term one at a time use the chain rule for terms containing y For we get Now solve for dy/dx

Differentiate Then gives us We can replace the y in the results with the explicit value of y as needed This gives us the slope on the curve for any legal value of x

Guidelines for Implicit Differentiation

Slope of a Tangent Line Substitute x = 3, y = -2 Given x3 + y3 = y + 21 find the slope of the tangent at (3,-2) 3x2 +3y2y’ = y’ Solve for y’ Substitute x = 3, y = -2

Using inside out Using product rule

Second Derivative Given x2 –y2 = 49 y’ =?? y’’ = Substitute