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, we start with the definition of the derivative and apply it to the function f(x) = p(x) - q(x). By expanding and rearranging, we can split the limit into separate limits which are just the derivatives of p(x) and q(x), respectively. Therefore, the derivative of f(x) is p'(x) - q'(x).

Step-by-step explanation:

To prove the Difference Rule for differentiation, we start with the definition of the derivative:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

Now, let's consider the function f(x) = p(x) - q(x), where p(x) and q(x) are differentiable functions. We can substitute this into the derivative definition:

f'(x) = lim(h->0) [(p(x + h) - q(x + h)) - (p(x) - q(x))] / h

Expanding and rearranging, we get:

f'(x) = lim(h->0) [p(x + h) - p(x) - q(x + h) + q(x)] / h

Using the limit properties and the definition of the derivative, we can split the limit into four separate limits:

f'(x) = lim(h->0) [p(x + h) - p(x)] / h - lim(h->0) [q(x + h) - q(x)] / h

These limits are just the derivatives of p(x) and q(x), respectively. So, we have:

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