Question

Express the product and quotient rules in Leibniz/ operator notation.

Short answer

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

Quotient Rule

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(\frac{g\left(x\right)}{h\left(x\right)}\right) \\ \frac{d}{dx}f\left(x\right)=\frac{\frac{d}{dx}g\left(x\right)*h\left(x\right)-\frac{d}{dx}h\left(x\right)*g\left(x\right)}{(h{\left(x\right))}^2} \end{gathered}$$

Step-by-step solution

Given Information 

Express the product and quotient rules in Leibniz/ operator notation.

Step 2:  Derivative 

Lets consider product of functions

f(x) = g(x)*h(x)

Express the product operator notation

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

Derivative 

Lets consider quotient of functions

f(x) = g(x)/h(x)

Express the product operator notation

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(\frac{g\left(x\right)}{h\left(x\right)}\right) \\ \frac{d}{dx}f\left(x\right)=\frac{\frac{d}{dx}g\left(x\right)*h\left(x\right)-\frac{d}{dx}h\left(x\right)*g\left(x\right)}{(h{\left(x\right))}^2} \\ f\text{'}\left(x\right)=\frac{g\text{'}\left(x\right)h\left(x\right)-h\text{'}\left(x\right)g\left(x\right)}{h{\left(x\right)}^2} \end{gathered}$$