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If a car speedometer reads 45 mph, that is not an average over the whole trip β€” it is your exact speed at that single split-second.
To find the instantaneous rate of change for any curve, we use the derivative from first principles.

The Limit Definition

The gradient of a line between two points (x,f(x))(x, f(x)) and (x+h,f(x+h))(x+h, f(x+h)) is Ξ”yΞ”x=f(x+h)βˆ’f(x)h\frac{\Delta y}{\Delta x} = \frac{f(x+h) - f(x)}{h}, where hh is a small horizontal step.
The derivative fβ€²(x)f'(x) is the gradient of the tangent line obtained by taking the limit as the step size hh shrinks to zero: fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
In the late 1600s, Isaac Newton and Gottfried Wilhelm Leibniz independently discovered this method to calculate instantaneous velocity and planetary orbits.
What happens when we apply this formal machinery to a line that already has a constant slope?

Differentiating a Linear Function

Let's differentiate the linear function f(x)=4x+7f(x) = 4x + 7 using first principles. We first evaluate the function at the shifted point x+hx+h: f(x+h)=4(x+h)+7=4x+4h+7f(x+h) = 4(x+h) + 7 = 4x + 4h + 7
Next, substitute f(x+h)f(x+h) and f(x)f(x) into the difference quotient: fβ€²(x)=lim⁑hβ†’0(4x+4h+7)βˆ’(4x+7)hf'(x) = \lim_{h \to 0} \frac{(4x + 4h + 7) - (4x + 7)}{h}
The terms 4x4x and 77 subtract to zero, leaving lim⁑hβ†’04hh\lim_{h \to 0} \frac{4h}{h}. Because hh approaches zero without being equal to zero, we cancel hh to get fβ€²(x)=4f'(x) = 4.