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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).

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.3: Implicit Differentiation
Problem 44E
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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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