Question

Prove the difference rule in two ways

a) using definition of the derivative

b) using sum and constant multiple rules

Short answer

We prove the difference rules using the given rules

Step-by-step solution

Given information

We are given two function f anf g

Prove the difference rule using the definition of the derivative

We get,

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

Proof using sum and constant times rule

We have,

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

Which can also be written as

$$\begin{gathered} (f+(-g\left)\right)\text{'}\left(x\right) \\ f\text{'}\left(x\right)+(-g\text{'}(x\left)\right)(Sumrule) \\ f\text{'}\left(x\right)-g\text{'}\left(x\right)(cons\tan trule) \end{gathered}$$