If `f(x)` is differentiable at `x=a` , find `(lim)_(x->a)(x^2f(a)-a^2\ f(x))/(x-a)` .

If f(x) is differentiable at x=c , then write the value of (lim)_(x->c)f(x) .

Show that f(x)=x^2 is differentiable at x=1 and find f^(prime)(1) .

If f(2)=4 and f^(prime)(2)=1 , then find (lim)_(x->2)(x\ f(2)-2f\ (x))/(x-2) .

If f(x) + f(y) = f(xy-sqrt(1-x^(2))sqrt(1-y^(2))), f(0) = (pi)/(2) and f is differentiable in (-1, 1). Then |lim_(x rarr 0)(2f(x)-pi)/(x)|=

If f (x) is a thrice differentiable function such that lim _(xto0)(f (4x) -3 f(3x) +3f (2x) -f (x))/(x ^(3))=12 then the vlaue of f '(0) equais to :

If f (x) is a thrice differentiable function such that lim _(xto0)(f (4x) -3 f(3x) +3f (2x) -f (x))/(x ^(3))=12 then the vlaue of f '''(0) equais to :

If f(x) is a twice differentiable function such that f'' (x) =-f(x),f'(x)=g(x),h(x)=f^2(x)+g^2(x) and h(10)=10 , then h (5) is equal to

A function f(x) having the following properties, (i) f(x) is continuous except at x=3 (ii) f(x) is differentiable except at x=-2 and x=3 (iii) f(0) =0 lim_(x to 3) f(x) to - oo lim_(x to oo) f(x) =3 , lim_(x to oo) f(x)=0 (iv) f'(x) gt 0 AA in (-oo, -2) uu (3,oo) " and " f'(x) le 0 AA x in (-2,3) (v) f''(x) gt 0 AA x in (-oo,-2) uu (-2,0)" and "f''(x) lt 0 AA x in (0,3) uu(3,oo) Then answer the following questions Find the Maximum possible number of solutions of f(x)=|x|

Find function f(x) which is differentiable and satisfy the relation f(x+y)=f(x)+f(y)+(e^(x)-1)(e^(y)-1)AA x, y in R, and f'(0)=2.

Find function f(x) which is differentiable and satisfy the relation f(x+y)=f(x)+f(y)+(e^(x)-1)(e^(y)-1)AA x, y in R, and f'(0)=2.