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Let f RR and F: R" → Rm be differentiable at p = R". Prove that the product fF: Rn → Rm is also differentiable at p. Verify that the differential at p is given by D(ƒF)(p)(h) = F(p)Df(p)(h) + f(p)DF(p)(h).

Let f RR and F: R" → Rm be differentiable at p = R". Prove that the product fF: Rn → Rm is also differentiable at p. Verify that the differential at p is given by D(ƒF)(p)(h) = F(p)Df(p)(h) + f(p)DF(p)(h).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I have the following question from multivariable analysis, please explain step by step on how to approach this, thank you in advance:

 

Let f RR and F : R" → Rm be differentiable at p € R". Prove that the product fF: R" → Rm is also differentiable at p. Verify that the differential at p is given by D(ƒF)(p)(h) = F(p)Dƒ(p)(h) + f(p)DF(p)(h).
Transcribed Image Text:Let f RR and F : R" → Rm be differentiable at p € R". Prove that the product fF: R" → Rm is also differentiable at p. Verify that the differential at p is given by D(ƒF)(p)(h) = F(p)Dƒ(p)(h) + f(p)DF(p)(h).
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