Implicit Differentiation

1. Implicit Differentiation • The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. • For example, , or y =x sin x,or in general y = f(x). However, some functions are defined implicitly. Some examples of implicit functions are: x2+ y2 = 25 x3+ y3 = 6xy

2. In some cases, it is possible to solve such an equation for y as an explicit function (or several functions) of x. The graphs of and (x) are the upper and lower semicircles of the circle x2+ y2 = 25.

3. Nonetheless, x3 + y3 = 6xyis the equation of a curve called the folium of Descartes shown here and it implicitly defines y as several functions of x. • It’s not easy to solve equation x3 + y3 = 6xy • for y explicitly as a function of x by hand. • A computer algebra system has no trouble. • However, the expressions it obtains are very complicated Fortunately, we don’t need to solve an equation for y in terms of x to find the derivative of y. • Instead, we can use the method of implicit differentiation. • This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’. • In the examples, it is always assumed that the given equation determines y implicitly as a differentiable function of x so that the method of implicit differentiation can be applied.

4. example: a. If x2 + y2 = 25, find . b. Find an equation of the tangent to the circle x2 + y2 = 25 at the point (3, 4). a. Differentiate both sides of the equation x2+ y2 = 25: Chain rule: At the point (3, 4) we have x = 3 and y = 4. • Thus, an equation of the tangent to the circle at (3, 4) is: • y– 4 = – ¾(x – 3) or 3x+ 4y = 25.

5. example: a. Find y’ if x3 + y3 = 6xy. b. Find the tangent to the folium of Descartes x3 + y3 = 6xy at the point (3, 3). c. At what points in the first quadrant is the tangent line horizontal? Differentiating both sides of x3 + y3 = 6xywith respect to x, regarding y as a function of x, and using the Chain Rule on y3 and the Product Rule on 6xy, we get: 3x2 + 3y2y’ = 6xy’ + 6y or x2 + y2y’ = 2xy’ + 2y Now, we solve for y’: When x = y = 3, y’ = – 1 • So, an equation of the tangent to the folium at (3, 3) is:y – 3 = – 1(x – 3) or x + y = 6.

6. The tangent line is horizontal if y’ = 0. • Using the expression for y’ from (a), we see that y’ = 0 when 2y – x2 = 0 (provided that y2 – 2x ≠ 0). • Substituting y = ½x2 in the equation of the curve, we get • x3+ (½x2)3 = 6x(½x2) • which simplifies to x6= 16x3. Since x ≠ 0 in the first quadrant, we have x3 = 16. If x = 161/3 = 24/3, then y = ½(28/3) = 25/3. • Thus, the tangent is horizontal at (0, 0) and • at (24/3, 25/3), which is approximately • (2.5198, 3.1748).

7. This can’t be solved for y.

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

9. tangent: normal:

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

11. example: • Find y” if x4+ y4 = 16. • Differentiating the equation implicitly with respect to x, we get • 4x3+ 4y3y’ = 0. To find y’’, we differentiate this expression for y’ using the Quotient Rule and remembering that y is a function of x:

12. However, the values of x and y must satisfy the original equation x4 + y4 = 16. So, the answer looks quite simple: