Derivative of even function is

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as f'(a^-)=lim_(hrarr0^(+))(f(a)-f(a-h))/(h) =lim_(hrarr0^(+))(f(a+h)-f(a))/(h) andf'(a^(+))=lim_(hrarr0^(+))(f(a+h)-f(a))/(h) =lim_(hrarr0^(+))(f(a)-f(a+h))/(h) =lim_(hrarr0^(+)) (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. If f is even function, which of the following is right hand derivative of f' at x=a?

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as f'(a^-)=lim_(hrarr0^(+))(f(a)-f(a-h))/(h) =lim_(hrarr0^(+))(f(a+h)-f(a))/(h) andf'(a^(+))=lim_(hrarr0^(+))(f(a+h)-f(a))/(h) =lim_(hrarr0^(+))(f(a)-f(a+h))/(h) =lim_(hrarr0^(+)) (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. If f is even function, which of the following is right hand derivative of f' at x=a?

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 ,

The derivative of an even function is :

The derivative of an even function is always an odd function.

The derivative of an even function is always an odd function.

The derivative of an even function is always an odd function.

Prove that the derivative of an even function is an odd function and that of an odd function is an even function.

Prove that the derivative of an even function is an odd function and that of an odd function is an even function.