derivative?

Question

why is the derivative of an odd function is always even function? can someone explain this.

Answer 1

If the function is odd, then the rate of change of the two sides are the same, the shape is the same, the only difference is that the two sides are upside down from each other. When you derive the function, it doesn't matter whether its upside down or not, all that matters is the rate of change, therefore, the derivative is always is even as the rate of change is the same on both sides.

Answer 2

Easy, odd functions normally have an odd exponent...like x^3...when you take the derivative you get 3x^2 and that's an even function. Again, notice the exponent. When exponents are even they reflect across the y-axis and are symmetrical unless other parts of the equation alter it's basic shape.

Ex. f(x)=x^2 and x=2 f(x)=4...and if x= -2 you get f(x)=4 as well. When you graph x^2 you have a symmetrical parabola, correct? Same with x^4, x^6 and so forth.

Does that answer your question?

Answer 3

derivative of x^n is nx^(n-1)
thus the degree of the derivative is always one less than that
of the function.
odd no minus 1 always gives an even number