Q. 88 - Calculus

Question

Prove the difference rule in two ways

a) using definition of the derivative

b) using sum and constant multiple rules

Step-by-Step Solution

Answer

We prove the difference rules using the given rules

Step 1: Given information

We are given two function f anf g

Step 2: Prove the difference rule using the definition of the derivative

We get,

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{(f-g)(x+h)-(f-g)\left(x\right)}{h} \\ \underset{h\to 0}{\lim }\frac{f(x+h)-g(x+h)-f\left(x\right)+g\left(x\right)}{h} \\ \underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}-\underset{h\to 0}{\lim }\frac{g(x+h)-g\left(x\right)}{h} \\ =f\text{'}\left(x\right)-g\text{'}\left(x\right) \\ Hence proved \end{gathered}$$

Step 2: Proof using sum and constant times rule

We have,

$$\begin{gathered} We are given \\ (f-g)\text{'}\left(x\right) \end{gathered}$$

Which can also be written as

$$\begin{gathered} (f+(-g\left)\right)\text{'}\left(x\right) \\ f\text{'}\left(x\right)+(-g\text{'}(x\left)\right) (Sum rule) \\ f\text{'}\left(x\right)-g\text{'}\left(x\right) (cons\tan t rule) \end{gathered}$$