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Section 14.4 Chain Rules with two variables

(3/23/08) Section Rules with two variablesOverview:In this Section we discuss procedures for differentiating composite functions with two vari-ables. Then we consider second-order and higher-order derivatives of such : Using the Chain Rule for one variable The general Chain Rule with two variables Higher order partial derivativesUsing the Chain Rule for one variablePartial derivatives of composite functions of the formsz=F(g(x, y)) can be found directly with theChain Rule for one variable , as is illustrated in the following three 1 Find thex-andy-derivatives ofz= (x2y3+ sinx) find thex-derivative, we consideryto be constant and apply the one- variable ChainRule formuladdx(f10) = 10f9dfdxfrom Section We obtain x[(x2y3+ sinx)10] = 10(x2y3+ sinx)9 x(x2y3+ sinx)= 10(x2y3+ sinx)9(2xy3+ cosx).

The general Chain Rule with two variables We the following general Chain Rule is needed to find derivatives of composite functions in the form z = f(x(t),y(t)) or z = f (x(s,t),y(s,t)) in cases where the outer function f has only a letter name.

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Transcription of Section 14.4 Chain Rules with two variables

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