Theorem 14. Properties of the Gradient Let f and g both be real-valued functions of two or three variables that are differentiable on a common open set O. 1. Sum Rule 2. Constant Multiple Rule 3. Product Rule v(f+g) = Vf+Vg (cf)=cf for all ceR v(fg) =fvg+gVf

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Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
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Prove parts 1, 2, and 3 of Theorem 14.

Theorem 14. Properties of the Gradient
Let f and g both be real-valued functions of two or three variables that are differentiable on a
common open set O.
1. Sum Rule
2. Constant Multiple Rule
3. Product Rule
v(f+g) = Vf+Vg
(cf)=cf for all ceR
v(fg) =fvg+gVf
4. Chain Rule If h is a differentiable real-valued function of one variable defined on a
domain D, and if the range of f satisfies R, D = Dw, then
v(hof) = (h' of)vf
Transcribed Image Text:Theorem 14. Properties of the Gradient Let f and g both be real-valued functions of two or three variables that are differentiable on a common open set O. 1. Sum Rule 2. Constant Multiple Rule 3. Product Rule v(f+g) = Vf+Vg (cf)=cf for all ceR v(fg) =fvg+gVf 4. Chain Rule If h is a differentiable real-valued function of one variable defined on a domain D, and if the range of f satisfies R, D = Dw, then v(hof) = (h' of)vf
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