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Kalkulus II,[object Object],Teguh Budi P, M.Si 	,[object Object],Sesion#25-26,[object Object],JurusanFisika,[object Object],FakultasMatematikadanIlmuPengetahuanAlam,[object Object]
Partial Derivatives,[object Object],The Chainrule,[object Object],2,[object Object],Outline,[object Object],1/9/2011,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Multivariable Functions and Their Derivatives(part 3),[object Object],3,[object Object],1/9/2011,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
z,[object Object],100,[object Object],An Introduction to Partial Derivatives,[object Object],10,[object Object],y,[object Object],10,[object Object],x,[object Object],1/9/2011,[object Object],4,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
When we have functions with more than one variable, we can find partial derivatives by holding all the variables but one constant.,[object Object],z,[object Object],100,[object Object],10,[object Object],y,[object Object],10,[object Object],x,[object Object],(eff sub ecks),[object Object],Note:,[object Object],is also written as,[object Object],1/9/2011,[object Object],5,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
z,[object Object],would give you the slope of the  tangent in the plane y=0 or in any plane with constant y.,[object Object],100,[object Object],10,[object Object],y,[object Object],10,[object Object],x,[object Object],In other words, how is changing one variable going to change the value of the function?,[object Object],1/9/2011,[object Object],6,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Mixed variables are also possible:,[object Object],Both answers are the same!,[object Object],1/9/2011,[object Object],7,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
The Chain Rule,[object Object],Consider a simple composite function:,[object Object],1/9/2011,[object Object],8,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
and another:,[object Object],1/9/2011,[object Object],9,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
This pattern is called the chain rule.,[object Object],and one more:,[object Object],1/9/2011,[object Object],10,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
If            is the composite of                 and                  , then:,[object Object],Find:,[object Object],example:,[object Object],Chain Rule:,[object Object],1/9/2011,[object Object],11,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
We could also do it this way:,[object Object],1/9/2011,[object Object],12,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Here is a faster way to find the derivative:,[object Object],Differentiate the outside function...,[object Object],…then the inside function,[object Object],1/9/2011,[object Object],13,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Another example:,[object Object],It looks like we need to use the chain rule again!,[object Object],derivative of the,[object Object],outside function,[object Object],derivative of the,[object Object],inside function,[object Object],1/9/2011,[object Object],14,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
The chain rule can be used more than once.,[object Object],Another example:,[object Object],(That’s what makes the “chain” in the “chain rule”!),[object Object],1/9/2011,[object Object],15,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
The formulas on the memorization sheet are written with      instead of      .   Don’t forget to include the      term!,[object Object],Derivative formulas include the chain rule!,[object Object],etcetera…,[object Object],1/9/2011,[object Object],16,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
The most common mistake on the chapter 3 test is to forget to use the chain rule.,[object Object],Every derivative problem could be thought of as a chain-rule problem:,[object Object],The derivative of x is one.,[object Object],derivative of outside function,[object Object],derivative of inside function,[object Object],1/9/2011,[object Object],17,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Divide both sides by,[object Object],The slope of a parametrized curve is given by:,[object Object],The chain rule enables us to find the slope of parametrically defined curves:,[object Object],1/9/2011,[object Object],18,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Example:,[object Object],These are the equations for an ellipse.,[object Object],1/9/2011,[object Object],19,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Thank You,[object Object],1/9/2011,[object Object],20,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]

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