1) Find the hundredth derivative of
\[
p(x)=\left(x+x^{5}+x^{7}\right)^{10}\left(1+x^{2}\right)^{11}\left(x^{3}+x^{5}+x^{7}\right) \text {. }
\]

Justify your answer.

Step 1 ：The hundredth derivative of a function can be quite complex to compute directly. However, we can observe that the function is a product of three functions, each raised to a power. This suggests that we could potentially use the product rule and the chain rule of differentiation to simplify the problem. However, the powers are quite high (10, 11, and 1 respectively), and the hundredth derivative would involve a lot of terms if we were to use the product rule directly.

Step 2 ：Another observation is that each of the three functions being multiplied together is a polynomial, and the derivative of a polynomial is another polynomial, with the degree reduced by one. This means that after a certain number of derivatives, the derivative of the polynomial will be zero.

Step 3 ：For the first function, \((x+x^{5}+x^{7})^{10}\), the highest power of x is 7, so after 7 derivatives, the derivative will be zero. Similarly, for the second function, \((1+x^{2})^{11}\), after 2 derivatives, the derivative will be zero. For the third function, \((x^{3}+x^{5}+x^{7})\), after 7 derivatives, the derivative will be zero.

Step 4 ：Therefore, after 7+2+7=16 derivatives, the derivative of the entire function will be zero. Since 100 is greater than 16, the hundredth derivative of the function will be zero.

Step 5 ：Final Answer: The hundredth derivative of the function is \(\boxed{0}\).