سؤال (باللغة الإنجليزية) عن الدوال !

[SSSSMMMM]

prove that if F:R--->R is an even function(that is F(-x)=F(x) for all x in R)
and has aderivative at every point,then the derivative is an odd function(that is F(-x)=-F(x) for all x in r)


also prove that if g:R---->R is adifferentiable odd function ,then g prime is an even function

[SSSSMMMM]

اين الصعوبة في السؤال سأكتب انا الحل في الايام القادمة قريبا ان شاء الله

[prime]

اتصور انه توجد أكثر من طريقة ولكن أسهل الطرق كالتالي :
------ ( @) We have : f (-x)=f(x) for all x in R
then d (f(x)/dx = f'(x) for all x and
d(f(-x)/dx = - f'(-x) for all x this by chain rule ( قاعدة التسلسل )
from ( @) : then f'(-x)=- f'(x) for all x
Q.E.D