If `f(x)=x|x|,` then prove that `f^(prime)(x)=2|x|`

If f(x)=|x+a^2a b a c a b x+b^2b c a c b c x+c^2|,t h e n prove that f^(prime)(x)=3x^2+2x(a^2+b^2+c^2)dot

If f (x) = cot x, then prove that : f(-x)=-f(x)

If (x)=tan x , the prove that :f(x)+f(-x)=0

If f(x)=x+(1)/(x) , then prove that : {f(x)}^(3)=f(x^(3))+3*f((1)/(x))

If f (x) =(x-1)/(x+1), then prove that f{f(x)}=-1/x.

If f(x)=(1)/(1-x) , then prove that : f[f{f(x)}]=x

If f(x)=x-(1)/(x) then prove that f(x)=-f((1)/(x))

If f(x)="log"(1+x)/(1-x) , then prove that: f((2x)/(1+x^(2)))=2f(x)

A function f:R rarr R satisfies the equation f(x+y)=f(x)f(y) for allx,y in R and f(x)!=0 for all x in R .If f(x) is differentiable at x=0 .If f(x)=2, then prove that f'(x)=2f(x) .

If f(x)=log_(e)x, then prove that :f(xyz)=f(x)+f(y)+f(z)