Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?

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Presentation transcript:

Implicit Differentiation Section 3.7a

Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?

We use implicit differentiation, so named b/c the functions are defined implicitly (hidden) within the original equation

Implicit Differentiation Process  We treat y as a differentiable function of x!!! 1.Differentiate both sides of the equation with respect to x. 2. Collect the terms with dy/dx on one side of the equation. 3. Factor out dy/dx. 4. Solve for dy/dx. Because of the Chain Rule, any time differentiating any term containing a y, also multiply by dy/dx!!!

Initial Guided Practice Find dy/dx: Does this answer make sense graphically? Implicit Differentiation: Solve for the derivative:

Initial Guided Practice Find the slope of the circle at First, find the slope of any point on the circle via implicit differentiation: Slope at the given point: Again, verify this answer graphically!

Initial Guided Practice Show that the slope dy/dx is defined at every point on the graph of Imp Diff: This formula for dy/dx is defined at every point (x,y) except for those points at which cos(y)=2… Which never happens!!!

We can use all of this new info to expand the POWER of the POWER RULE!!!  A proof: First, let p and q be integers with q > 0 and suppose that: Imp. Diff.! Subst. for y Law of Exp. Final Answer!

The EXPANDED Power Rule: If n is any rational number, then (If n < 1, then the derivative does not exist at x = 0) WHY NOT??? Note: Before, the power rule worked for integers only…  now, it works for any rational number power (i.e., fraction) What is the derivative of the square root function???

Additional Guided Practice For each of the following, find dy/dx.

Additional Guided Practice For each of the following, find dy/dx.

Additional Guided Practice For each of the following, find dy/dx.

Additional Guided Practice For each of the following, find dy/dx.

Additional Guided Practice For each of the following, find dy/dx.