Question

is the derivative of a polynomial is always a polynomial?

Answer 1

Yes, because of the power rule.

Answer 2

Wikipedia defines polynomial to be:

A polynomial is either zero or can be written as the sum of a finite number of non-zero terms.

So if you have:
\[\large {d\over dx} (a_nx^n + a_{n-1}x^{n-1} + ... +a_2x^2 + a_1x + a_0) \]
You will get
\[\large {d\over dx} (a_nx^n) + {d\over dx}(a_{n-1}x^{n-1}) + ... +{d\over dx}(a_2x^2 ) +{d\over dx}( a_1x) +{d\over dx}( a_0) \]

We know that any of those terms will have a derivative that is also a term:
\[\large {d\over dx}ax^b\]
Is equal to
\[\large abx^{b-1}\]...which is also a term.

Answer 3

thanks