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

the value of lim_(h to 0) (f(x+h)+f(x-h))/h is equal to

If f(x)=(1)/(x), evaluate lim_(h rarr0)(f(x+h)-f(x))/(h)

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as f'(a^-)=lim_(h to 0^(+))(f(a)-f(a-h))/(h) =lim_(hto0^(+))(f(a+h)-f(a))/(h) andf'(a^(+))=lim_(h to 0^+)(f(a+h)-f(a))/(h) =lim_(hto0^(+))(f(a)-f(a+h))/(h) =lim_(hto0^(+))(f(a)-f(x))/(a-x) respectively. Let f be a twice differentiable function. We also know that derivative of a even function is odd function and derivative of an odd function is even function. The statement lim_(hto0)(f(-x)-f(-x-h))/(h)=lim_(hto0)(f(x)-f(x-h))/(-h) implies that for all x"inR ,

If f is an even function such that lim_(h rarr0)(f(h)-f(0))/(h) has some finite non-zero value,then prove that f(x) is not differentiable at x=0.

(1) If f(x) = sqrt(x) , prove that : (f(x + h) - f(x))/(h) = (1)/(sqrt(x + h) + sqrtx) . (2) If f(x) = x^(2) , find : (f(1.1) - f(1))/(1.1 - 1) .

Let the derivative of f(x) be defined as D*f(x)=lim_(h rarr0)(f^(2)(x+h)-f^(2)(x))/(h) where f^(2)(x)=(f(x))^(2) if u=f(x),v=g(x), then the value of D*{u.v} is

let f(x) be differentiable at x=h then lim_(x rarr h)((x+h)f(x)-2hf(h))/(x-h) is equal to

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