Question

Following the proof of the Sum Rule for differentiation, prove the Difference Rule for differentiation. That is, prove that if f(x)=p(x)−q(x), then f′(x)=p′(x)−q′(x).

Answer 1

To prove the Difference Rule for differentiation, which states that the derivative of the difference of functions is the difference of their derivatives, one applies the limit definition of the derivative to the function represented as f(x) = p(x) - q(x), resulting in the derivation of f'(x) = p'(x) - q'(x).

Step-by-step explanation:

To prove the Difference Rule for differentiation, we start with the assumption that the function f(x) can be expressed as the difference of two functions p(x) and q(x), in other words, f(x) = p(x) − q(x). By definition of the derivative, the derivative of f at point x, denoted f′(x), is the limit of the difference quotient as h approaches 0:

f'(x) = lim_(h→0) [(f(x+h) - f(x)) / h]

Substituting the expression for f(x), we have:

f'(x) = lim_(h→0) [(p(x+h) - q(x+h) - (p(x) - q(x))) / h]

Using properties of limits and rearranging terms, we can break this down into:

f'(x) = lim_(h→0) [(p(x+h) - p(x)) / h] - lim_(h→0) [(q(x+h) - q(x)) / h]

These limits are precisely the definitions of p′(x) and q′(x), respectively:

f'(x) = p'(x) - q'((x))

And there we have it: the proof of the Difference Rule states that the derivative of the difference of two functions is equal to the difference of their derivatives.