Question

Express the constant multiple, sum, and difference rules in Leibniz/operator notation

Short answer

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}k \\ \frac{d}{dx}f\left(x\right)=\frac{d}{dx}g\left(x\right)+\frac{d}{dx}h\left(x\right) \\ \frac{d}{dx}f\left(x\right)=\frac{d}{dx}g\left(x\right)-\frac{d}{dx}h\left(x\right) \end{gathered}$$

Step-by-step solution

Given Information 

Express the constant multiple, sum, and difference rules in Leibniz/operator notation

Constant

Consider a constant function

$$\begin{gathered} f\left(x\right)=k \\ weneedtoexpressthedifferentiationinoperatornotation \\ \frac{d}{dx}f\left(x\right)=\frac{d}{dx}k \\ f\text{'}\left(x\right)=0 \end{gathered}$$

Sum

Consider a sum function

$f\left(x\right)=g\left(x\right)+h\left(x\right)$

express the differentiation in operator notation

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}g\left(x\right)+\frac{d}{dx}h\left(x\right) \\ f\text{'}\left(x\right)=g\text{'}\left(x\right)+h\text{'}\left(x\right) \end{gathered}$$

Difference

Consider difference of functions

$f\left(x\right)=g\left(x\right)-h\left(x\right)$

Express it in operator notation

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}g\left(x\right)-\frac{d}{dx}h\left(x\right) \\ f\text{'}\left(x\right)=g\text{'}\left(x\right)-h\text{'}\left(x\right) \end{gathered}$$