To find the derivative of the function g(x) = 8x using the definition of the derivative, we start by applying the limit definition:
Definition of the Derivative: The derivative of a function g(x) at a point x is defined as:
g'(x) = lim (h → 0) [(g(x + h) – g(x)) / h]
Now, let’s substitute our function into this definition:
1. Calculate g(x + h):
Since g(x) = 8x, we have:
g(x + h) = 8(x + h) = 8x + 8h
2. Substitute into the limit definition:
g'(x) = lim (h → 0) [(8x + 8h – 8x) / h]
3. Simplify the expression:
g'(x) = lim (h → 0) [8h / h]
4. Cancel h (as long as h ≠ 0):
g'(x) = lim (h → 0) [8]
5. Now, taking the limit as h approaches 0:
g'(x) = 8
Thus, the derivative of the function g(x) = 8x is:
g'(x) = 8
This result tells us that the slope of the function at any point x is constant and equal to 8, meaning the function is linear with a consistent rate of change.