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

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

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?

Statement I Integral of an even function is not always an odd function. Statement II Integral of an odd function is an even function .

Which of the following function is an even function ?

Show that any function can be expressed as the sum of an odd function and even function

Which of the following functions is an odd function:

Which of the following functions is an odd function?

Which of the following functions is an odd functions ?

Which of the following is not an odd function ?

The function f(x)=log(x+sqrt(x^(2)+1)) , is (a) an even function (b) an odd function (c ) a periodic function (d) Neither an even nor an odd function.