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Mathematics LibreTexts

6.3: Linear Differential Operators

Your calculus class became much easier when you stopped using the limit definition of the derivative, learned the power rule, and started using linearity of the derivative operator.

Example 64

Let \(V\) be the vector space of polynomials of degree 2 or less with standard addition and scalar multiplication.

\[V = \{a_{0}\cdot1 + a_{1}x + a_{2} x^{2} | a_{0},a_{1},a_{2} \in \Re \}\]

Let \(\frac{d}{dx} \colon V\rightarrow V\) be the derivative operator. The following three equations, along with linearity of the derivative operator, allow one to take the derivative of any 2nd degree polynomial:

$$
\frac{d}{dx} 1=0,~\frac{d}{dx}x=1,~\frac{d}{dx}x^{2}=2x\,. 
$$

In particular

$$
\frac{d}{dx} (a_{0}\cdot1 + a_{1}x + a_{2} x^{2}) = 
 a_{0}\frac{d}{dx}\cdot1 + a_{1} \frac{d}{dx} x + a_{2} \frac{d}{dx} x^{2}
 = 0+a_{1}+2a_{2}.
$$

Thus, the derivative acting any of the infinitely many second order polynomials is determined by its action for just three inputs.

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