`lim_(h->0) (f(2h+2+h^2)-f(2))/(f(h-h^2+1)-f(1))` given that `f'(2)=6 and f'(1)=4` does not exist (a) is equal to `- 3/2` (b) is equal to `3/2` (c) is equal to 3

Let 3f(x)-2f((1)/(x))=x the f'(2) is equal to

IF f(x) =2x^2 +bx +c and f(0) =3 and f (2) =1, then f(1) is equal to

Iff(x)=x+int_0^1t(x+t)f(t)dt ,t h e nt h ev a l u eof(23)/2f(0) is equal to _________

If f(x) =(3(Inx)^(2)+(In x)^(4))/(x(1+(In x)^(2)-(Inx)^(3))^(2))dx and f(1) = 0, then the value of |3f(e^(2))| is equal to

Prove that lim_(x->0) (f(x+h)+f(x-h)-2f(x))/h^2=f''(x) (without using L' Hospital srule).

Let f(x)=(lim)_(h->0)(("sin"(x+h))^(1n(x+h))-(sinx)^(1nx))/hdot Then f(pi/2) equal to (a)0 (b) equal to 1 (c)In pi/2 (d) non-existent

Let f be a differentiable function such that f(1) = 2 and f'(x) = f (x) for all x in R . If h(x)=f(f(x)), then h'(1) is equal to

If f(x)=2x+3 , g(x)=1-2x and h(x)=3x. Prove that f o(g o h) = (f o g ) o h

If f(x)=2x+3, g(x) = 1-2x and h(x)=3x . Prove that f o (g o h) = (f o g) o h

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these