Q4 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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 1: 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

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

Step 3: Derivative

Lets consider quotient of functions

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

Express the product   operator notation

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