MAT.126
2.5-2.6

2.5 Implicit Differentiation

Distinguish between functions written in implicit form and explicit form

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Explicit: y = f(x)Implicit: y and f(x) are mixed together

Use implicit differentiation to find the derivative of a function

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Differentiate both sides of the equation with respect to x.Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.Factor dy/dx out of the left side of the equation.Solve for dy/dx.

Chain Rule

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To differentiate an implicitly defined function one must use the chain rule on all terms involving y. d dy---[ f(y) ] = f'(y) ----dx dx

Process

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Differentiate both sides of the equation with respect to x.Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.Factor dy/dx out of the left side of the equation.Solve for dy/dx.

Graphing Implicit Relations

Slope of an Implicit Graph

Explicit Domains for Implicit Functions

Higher-Order Implicit Derivatives

Tangent Line to an Implicit Graph

2.6 Related Rates

Find a related rate

Use known equations

Differentiate implicitly wrt t

Use related rates to solve real-life problems

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GuidelinesIdentify all given quantities and quantities to be determined. Make a sketch and label the quantities.Write an equation involving the variables whose rates of change either are given or are to be determined.Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t.After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.