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3.3 Implicit Differentiation and Related Rates

3.3 Implicit Differentiation and Related Rates. Finding dy/dx by Implicit Differentiation Differentiate each term of the equation with respect to x, treating y as a function of x. Move all terms involving dy/dx to the left side of the equation and move the other terms to the right side.

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3.3 Implicit Differentiation and Related Rates

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  1. 3.3 Implicit Differentiation and Related Rates

  2. Finding dy/dx by Implicit Differentiation Differentiate each term of the equation with respect to x, treating y as a function of x. Move all terms involving dy/dx to the left side of the equation and move the other terms to the right side. Factor out dy/dx on the left side of the equation. Divide both sides of the equation by the factor that multiplies dy/dx.

  3. In implicit differentiation we differentiate an equation involving x and y, with y treated as a function of x. There are some applications where x and y are related by an equation, and both variables are functions of a third variable t. Often the formulas for x and y as functions of t are not known.

  4. When we differentiate such an equation with respect to t, we derive a relationship between rates of change dy/dt and dx/dt. We say that these derivatives are related rates. The equation relating the rates may be used to find one of the rates when the other is known.

  5. Problem • Suppose that x and y are both differentiable functions of t and are related by the equation x2 + 5y2 = 36. • Differentiate each term with respect to t, and show the resulting equation for dy/dt. • Calculate dy/dt at a time when x = 4, y = 2, and dx/dt = 5.

  6. Suggestions for Solving Related Rates Problems • Draw a picture, if possible. • Assign letters to quantities that vary, and identify one variable, say t, on which the other variables depend. • Find an equation that relates the variables to each other. • Differentiate the equation with respect to the independent variable t. Use the chain rule whenever appropriate. • Substitute all specified values for the variables and their derivatives. • Solve the derivative that gives the unknown rate.

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