Implicit Differentiation & Related Rates

Implicit and Explicit Functions Explicit equations are solved for y and written as a function of x. Example: y = x2 + 4x Implicit equations aren’t solved for y. Example: x2 + 4y = 24

How To Use Implicit Differentiation Differentiate both sides of the equation with respect to x. Apply the basic rules of differentiation. Any time you differentiate a y put a dy/dx next to it. Isolate all of the terms with a dy/dx on one side of the equation. Factor out the dy/dx. Divide on both sides to solve for dy/dx. Video

Finding The Slope Of A Graph At A Certain Point Follow the same rules of implicit differentiation. After finding the derivative of the equation, plug the point (x,y) into the equation and solve.

Finding The Tangent Line To The Graph Follow the same rules of implicit differentiation and find the slope at the point given. Use the point-slope formula by plugging in the point (x,y) and the slope. Point-Slope Formula: y – y1 = m(x – x1)

Related Rates If 2 variables both vary with respect to time and have a relation between them, it cam be expressed as a rate of change of one in terms of the other. Rate of Change of Q = dQ/dt

Setting Up Related Rates Identify the given quantities and the quantities that are to be determined. Use a sketch if necessary. Write an equation involving the variables whose rates of change are given or are to be determined.

Solving Related Rates Use implicit differentiation to differentiate both sides of the equation with respect to the quantity that is being found. Use givens to substitute into the resulting equation, then solve. Related Rates Video