Let $f(x)$ be a polynomial and $g(x) = f^{\prime}(x)$ be its derivative. If the degree of $(f(x) + f(-x))$ is 10, then the degree of $(g(x) - g(-x))$ is ________.

Any polynomial $f(x)$ can be decomposed into the sum of an even polynomial $f_e(x)$ and an odd polynomial $f_o(x)$:

Then:

Given that the degree of $(f(x) + f(-x)) = 2f_e(x)$ is $10$, the highest degree term of the even part $f_e(x)$ is of power $10$.

Now, $g(x) = f^{\prime}(x) = f_e^{\prime}(x) + f_o^{\prime}(x)$. Since the derivative of an even function is odd and the derivative of an odd function is even:

• $f_e^{\prime}(-x) = -f_e^{\prime}(x)$

• $f_o^{\prime}(-x) = f_o^{\prime}(x)$

Therefore:

Since $f_e(x)$ has degree $10$, its derivative $f_e^{\prime}(x)$ has degree:

Thus, the degree of $(g(x) - g(-x))$ is9.