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Chapter 3.7 in Calculus textbook May 2010

dy dx. Implicit Differentiation. Chapter 3.7 in Calculus textbook May 2010. xy 2. xy. 3x 4 y 3. x 3 y 2. x4y. 4x 2 y. x 2 y. 2x 4 y 2. y = 3x 5 – 7x is easy to differentiate. y 2 = 3x 5 – 7x is a little harder.

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Chapter 3.7 in Calculus textbook May 2010

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  1. dy dx Implicit Differentiation Chapter 3.7 in Calculus textbook May 2010 xy2 xy 3x4y3 x3y2 x4y 4x2y x2y 2x4y2

  2. y = 3x5 – 7x is easy to differentiate. y2 = 3x5 – 7x is a little harder. y2 + y = 3x5 – 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.

  3. dy dx dy dx dy dx dy dx dy dx dy dx dy dx Example 1: Find of y3 – 4y2 = x5 +3x4 Take the derivative of each side, and multiply each y by Factor Divide and Solve 3y2 - 8y = 5x4 + 12x3 (3y2 – 8y) = 5x4 + 12x3 5x4 + 12x3 3y2 – 8y =

  4. dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx Example 2: Find of 3x2 + 5xy2 – 4y3 = 8 Take the derivative of each side, and multiply each y by Collect like terms Factor Divide and Solve 6x + (10xy + 5y2) – 12y2 = 0 10xy – 12y2 = -6x – 5y2 (10xy – 12y2) = -6x – 5y2 -6x – 5y2 10xy – 12y2 (Product Rule of 5xy2) =

  5. dy dx dy dx dy dx dy dx dy dx dy dx dy dx dy dx Example 3: Find the tangent of x2 – xy + y2 = 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 Find Tangent =

  6. 14 5 4 5 4 5 dy dx Example 3: Find the tangent of x2 – xy + y2 = 7 at the point (-1, 2) 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 y – 2x 2y – x (2) – 2(-1) 4 2(2) – (-1) 5 y – 2 = (x – (-1)) y = x + = =

  7. x + y x – y • More Practice • Find the derivatives of: • x3 – y3 = y • 2) x2 – 16xy + y2 = 1 • 3) = 3 at (2, 1) Answers

  8. x + y x – y 3x2 1 + 3y2 8y – x y – 8x 1 2 • More Practice • Answers: • x3 – y3 = y • 2) x2 – 16xy + y2 = 1 • 3) = 3 at (2, 1)

  9. Works Consulted Finney, Ross, Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus. New Jersey: Pretince Hall, 2003. Print. Kahn, David S. Cracking the AP Calculus AB & BC Exams. New York: Random House, 2009. Print.

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