Is `f(x)xx g(x)xx r(x)=LCM[f(x),g(x),r(x)]xx GCD [f(x),g(x),r(x)]`?

If f_r(x),g_r(x),h_r(x),r=1,2,3 are polynomials such that f_r(a)=g_r(a)=h_r(a),r=1,2,3a n d F(x)=|[f_1(x),f_2(x),f_3(x)],[g_1(x),g_2(x),g_3(x)],[h_1(x),h_2(x),h_3(x)]| then F^(prime)(x)a tx=a is____________________

f(x)=(1+x) g(x)=(2x-1) Show that fo(g(x))=go(f(x))

Let F(x)=f(x)g(x)h(x) for all real x ,w h e r ef(x),g(x),a n dh(x) are differentiable functions. At some point x_0,F^(prime)(x_0)=21 F(x_0),f^(prime)(x_0)=4f(x_0),g^(prime)(x_0)=-7g(x_0), and h^(prime)(x_0)=kh(x_0) . Then k is________

If f(x),g(x)a n dh(x) are three polynomial of degree 2, then prove that varphi(x)=|f(x)g(x)h(x)f'(x)g'(x h '(x)f' '(x)g' '(x h ' '(x)| is a constant polynomial.

If f(x) =x^(2) -2,g (x) =2x+ 1 then f^(@) g (x)

Let f:R-> R and g:R-> R be respectively given by f(x) = |x| +1 and g(x) = x^2 + 1 . Define h:R-> R by h(x)={max{f(x), g(x)}, if xleq 0 and min{f(x), g(x)}, if x > 0 .The number of points at which h(x) is not differentiable is

If intsinx d(secx)=f(x)-g(x)+c ,t h e n f(x)=secx (b) f(x)=tanx g(x)=2x (d) g(x)=x

If f(x-y)=f(x).g(y)-f(y).g(x) and g(x-y)=g(x).g(y)+f(x).f(y) for all x in R . If right handed derivative at x=0 exists for f(x) find the derivative of g(x) at x =0

Let f(a)=g(a)=k and their nth derivatives exist and be not equal for some n. If lim_(xtoa) (f(a)g(x)-f(a)-g(a)f(x)+g(a))/(g(x)-f(x))=4 then find the value of k.

Suppose fa n dg are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all xa n df^(prime)a n dg' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x))e q u a l (a) (-2f^(prime)(x))/f (b) (2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))