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Differentiation
[Definition] [Differentiation Rules] [Implicit Differentiation] [Related Rates] [Differentiation Links]
Definition
- The derivative is found by using the limit used to define the slope of a tangent line. The derivative of f at x is give by
 (provided a limit exists)
- example: find the derivative of
Differentiation Rules
- Using the definition of the derivative can take a long time in finding the derivative of most functions. There are several rules that can be used to find the derivative in a faster method.
 
- example: find the derivative of
- example: find the derivative of
Implicit Differentiation
- So far, only explicit differentiation has been used to find the derivative of functions. In some cases, both x and y need to be differentated. In these cases, implicit differentiation is used. The steps for finding the derivative of inplicit functions are as follows:
- 1. Differentiate both sides of the equation with respect to x.
- 2. Collect all terms involving
 on the left side of the equation and move all other terms to the right side of the equation.
- 3. Factor out of the left side of the equation.
- 4. Solve for by dividing both sides of the equation by the left-hand factor that does not contain .
- example: differentiate
- 1. Differentiate both sides of the equation with respect to x.
- 2. Collect all terms involving on the left side of the equation and move all other terms to the right side of the equation.
- 3. Factor out of the left side of the equation.
- 4. Solve for by dividing both sides of the equation by the left-hand factor that does not contain .
Related Rates
- A real life application of differentiation is finding related rates. Finding the related rates is finding the rates of change of two or more related variables that are changing with respect to time. The guidelines for solving related rates problems are as follows:
- 1. Identify all given quantities and quantities to be determined. Make a sketch and label the quantities.
- 2. Write an equation involving the variables whose rates of change either are given or are to be determined.
- 3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t.
- 4. After completing step 3, substitue into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.
- example: let V be the volume of the balloon and let r be its radius with volume increasing at a rate of 3.25 inches per minute, or the rate of change of volume (dV/dt) 13/4 being
- find:
 when 
- To find the rate of change of the radius, you must find an equation that relates the radius r to the volume V.
 (volume of a sphere)
- Implicit differentiation with respect to t produces the following results:
 solve for dr/dt
- Finally, when , the rate of change of the radius can be found by plugging in the given as follows:
 inches per minute
Differentiation Links
Calculus Phobe  - differentiation videos
Visual Calculus: Tangent Lines - tangent line visualization/demonstration
Visual Calculus: Definition of a Deriviative at a Point - differentiation example
Visual Calculus Drill: Product Rule - product rule drill with answer explanations
Visual Calculus: Quiz on Differentiation - multiple choice quiz on differentiation
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