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Implicit Differentiation 3.6

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Implicit Differentiation So far, all the equations and functions we looked at were all stated explicitly in terms of one variable: In this function, y is defined explicitly in terms of x. If we re-wrote it as xy = 1, y is now defined implicitly in terms of x. It is easy to find the derivative of an explicit function, but what about:

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This is not a function, but it would still be nice to be able to find the slope. Do the same thing to both sides. Note use of chain rule.

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This can’t be solved for y. This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. 2 Solve for.

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Find dy/dx if:

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Chain Rule

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Find dy/dx if: Product Rule!

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We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for. Note product rule. Find the equations of the lines tangent and normal to the curve at.

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tangent: Find the equations of the lines tangent and normal to the curve at. normal: Normal line is perpendicular to tangent

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Find derivative at (1, 1) Product Rule is easier than quotient rule, so let’s cross multiply!

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Higher Order Derivatives Find if. Substitute back into the equation.

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