Chapter 3.7 in Calculus textbook May 2010 xy 4x 2 y x 3 y 2 3x 4 y 3 x4y xy 2 2x 4 y 2 x 2 y dy dx

y = 3x 5 – 7x is easy to differentiate. y 2 = 3x 5 – 7x is a little harder. y 2 + y = 3x 5 – 7x would be very hard to differentiate if it was not for implicit differentiation. When x and y are both functions in the same equation, implicit differentiation allows you to find the rate at which y changes as x changes, or dy/dx.

Take the derivative of each side, and multiply each y by Factor Divide and Solve Example 1: Find of y 3 – 4y 2 = x 5 +3x 4 dy dx dy dx 3y 2 - 8y = 5x x 3 (3y 2 – 8y) = 5x x 3 5x x 3 3y 2 – 8y dy dx dy dx dy dx dy dx dy dx =

Example 2: Find of 3x 2 + 5xy 2 – 4y 3 = 8 dy dx Take the derivative of each side, and multiply each y by Collect like terms Factor Divide and Solve 6x + (10xy + 5y 2 ) – 12y 2 = 0 10xy – 12y 2 = -6x – 5y 2 (10xy – 12y 2 ) = -6x – 5y 2 -6x – 5y 2 10xy – 12y 2 dy dx dy dx dy dx (Product Rule of 5xy 2 ) dy dx dy dx dy dx dy dx dy dx =

Example 3: Find the tangent of x 2 – xy + y 2 = 7 at the point (-1, 2) Take the derivative of each side, and multiply each y by Collect like terms Factor Divide to get derivative 2x – (x + y) + 2y = 0 2y – x = y – 2x (2y – x) = y – 2x y – 2x 2y – x dy dx dy dx dy dx dy dx dy dx dy dx dy dx = Find Tangent dy dx

Example 3: Find the tangent of x 2 – xy + y 2 = 7 at the point (-1, 2) y – 2x 2y – x (2) – 2(-1) 4 2(2) – (-1) 5 y – 2 = (x – (-1)) y = x + dy dx = Evaluate the derivative for x = -1 and y = 2 The derivative is the slope of the tangent and we are given the x, y coordinates =

More Practice Find the derivatives of: 1)x 3 – y 3 = y 2) x 2 – 16xy + y 2 = 1 3) = 3 at (2, 1) x + y x – y

More Practice Answers: 1)x 3 – y 3 = y 2) x 2 – 16xy + y 2 = 1 3) = 3 at (2, 1) 3x y 2 8y – x y – 8x 1212 x + y x – y

Works Consulted Finney, Ross, Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus. New Jersey: Pretince Hall, Print. Kahn, David S. Cracking the AP Calculus AB & BC Exams. New York: Random House, Print.