If y = f(x) is a differentiable function of x. Then-

If y = f(u) is a differentiable functions of u and u = g(x) is a differentiable functions of x such that the composite functions y = f[g(x) ] is a differentiable functions of x then (dy)/(dx) = (dy)/(du) . (du)/(dx) Hence find (dy)/(dx) if y = sin ^(2)x

If y =f(u) is differentiable function of u, and u=g(x) is a differentiable function of x, then prove that y= f [g(x)] is a differentiable function of x and (dy)/(dx)=(dy)/(du)xx(du)/(dx) .

If y=f(x) is a differentiable function x such that invrse function x=f^(-1) y exists, then prove that x is a differentiable function of y and (dx)/(dy)=(1)/((dy)/(dx)) where (dy)/(dx) ne0 Hence, find (d)/(dx)(tan^(-1)x) .

If f(x) is a differentiable function for all x gt 0 and lim_(y to x)(y^(2)f(x)-x^(2)f(y))/(y-x)=2 , then the value of f'(x) is -

Given that f(x) is a differentiable function of x and that f(x) ,f(y) = f(x) + f(y) +f(xy) -2 and that f(1)=2 , f(2) = 5 Then f(3) is equal to

Let y = f(x) be a differentiable function, AA x in R and satisfies, f(x) = x+ int_(0)^(1)x^(2)z f(z) dz + int_(0)^(1)x z^(2) f(z) dz , then

Suppose y=f (x) is differentiable functions of x on an interval I and y is one - one, onto and (dy) / (dx) ne 0 on I . Also if f^(-1)(y) is differentiable on f(I) then prove that (dx)/(dy) =(1)/(dy//dx), dy/dx ne 0

Let y=f(x) be a differentiable function AA x in R and satisfies: f(x)=x+int_(0)^(1)x^(2)zf(z)dz+int_(0)^(1)xz^(2)f(z)dz.

f(x+y)=f(x).f(y)AA x,y in R and f(x) is a differentiable function and f'(0)=1,f(x)!=0 for any x.Findf(x)