Differentiation is a fundamental concept in calculus that finds the instantaneous rate of change of a function at any given point. Here we see a parabola, and at the red point, the derivative gives us the slope of the tangent line, showing exactly how fast the function is changing at that specific location.
The derivative process uses the limit definition to find the instantaneous rate of change. For the function f of x equals x squared, the derivative is 2x. Watch how the slope of the tangent line changes as we move along the curve. At x equals 1, the slope is 2. At x equals 3, the slope is 6, showing the function is increasing much faster.
There are several fundamental derivative rules that make differentiation easier. The power rule states that the derivative of x to the n is n times x to the n minus 1. The constant rule says the derivative of any constant is zero. Here we see x cubed in blue and its derivative 3 x squared in red. Notice how at x equals 1, the original function has value 1, while its derivative has value 3, showing the rate of change at that point.
Differentiation has many real-world applications. In physics, if we have a position function, its derivative gives us velocity, and the derivative of velocity gives us acceleration. Here we see position s of t equals 16 t squared in blue, and its derivative, velocity v of t equals 32 t, in red. Watch how as time increases, both position and velocity increase, but velocity increases at a constant rate since acceleration is constant at 32 feet per second squared.
To summarize what we have learned about differentiation: It is the process of finding instantaneous rates of change. The derivative gives us the slope of the tangent line at any point on a curve. We have basic rules that make calculations easier, and differentiation has countless real-world applications from physics to economics. Understanding differentiation is fundamental for success in calculus and beyond.