Question

The general polynomial $P$ of degree $n$ in the variable $x$ has the form $P(x)=\sum_{k=0}^{n} a_{k} x^{k}=$ $a_{0}+a_{1} x+\ldots+a_{n} x^{n} .$ What is the derivative (with respect to $\left.x\right)$ of $P$ ?

Answer 1

The derivative of a polynomial \(P(x)\) of degree \(n\) with respect to \(x\) is obtained by differentiating each term in the polynomial with respect to \(x\). Using the power rule for differentiation, which states that the derivative of \(x^k\) with respect to \(x\) is \(k x^{k-1}\), the derivative of \(P(x)\) can be expressed as:
\[
P^{\prime}(x)=\sum_{k=1}^n k a_k x^{k-1}=a_1+2 a_2 x+3 a_3 x^2+\cdots+n a_n x^{n-1}
\]
Note that the constant term \(a_0\) is not included in the sum because its derivative with respect to \(x\) is 0 .