3.5 – Implicit Differentiation

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Presentation transcript:

3.5 – Implicit Differentiation

Try It If f(x) = (x7 + 3x5 – 2x2)10, determine y΄. Answer: f΄(x) =10(x7 + 3x5 – 2x2)9(7x6 + 15x4 – 4x) Now write the answer above only in terms of y if y = (x7 + 3x5 – 2x2). Answer: f ΄(x) = d[y10]/dx = 10y9y΄

Try It If y is some unknown function of x, determine d[y35] / dx. Answer: d[y35] / dx =_____________ In general, if y is some function of x, then what is d[yn] / dx? Answer: d[yn] / dx = ______________

Power Rule for Implicit Differentiation If y is some unknown function of x (or difficult to determine), then using the chain rule.

Purpose Which of these would be easy to solve for y and differentiate? 9x + x2 – 2y = 5 5x – 3xy + y2 = 2y Easy to solve for y and differentiate Not easy to solve for y and differentiate

The Idea In equations like 5x – 3xy + y2 = 2y, we simply assume that y = f(x), or some function of x which is not easy to find. Process wise, simply take the derivative of each side of the equation with respect to x and when we encounter a yn, we use

Examples Determine dy/dx for the following. 1. y3 = 2x 2. x2y3 = –7 3. y1/2 – 3x1/3y1/4 = 2x With products of yn and xn, you must use the product rule. Why? This is also true for quotients, but with the quotient rule.

You Try It Determine dy/dx for the following.

Derivative of Inverse Trig Functions

Derivative of Inverse Trig Functions

Derivative of Inverse Trig Functions