MAT.126 2.5-2.6

The text details the application of related rates and implicit differentiation in solving mathematical problems. It describes a structured approach to using related rates, emphasizing the importance of identifying all given and unknown quantities, creating a labeled sketch, and writing an appropriate equation that includes the variables whose rates of change are involved.

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MAT.126 2.5-2.6

2.6 Related Rates

Use related rates to solve real-life problems

Guidelines

Identify 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.

Find a related rate

Differentiate implicitly wrt t

Use known equations

2.5 Implicit Differentiation

Use implicit differentiation to find the derivative of a function

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.

Tangent Line to an Implicit Graph

Higher-Order Implicit Derivatives

Explicit Domains for Implicit Functions

Slope of an Implicit Graph

Graphing Implicit Relations

Process

Chain Rule

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

Distinguish between functions written in implicit form and explicit form

Explicit: y = f(x)

Implicit: y and f(x) are mixed together