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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 $P$ ?
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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 .
by Diamond (89,043 points)

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