Differentiation is a fundamental process in calculus that finds the instantaneous rate of change of a function. The derivative represents the slope of the tangent line at any point on the curve. Here we see a parabola and its tangent line at a specific point, showing how differentiation captures the rate of change.
The derivative is formally defined as the limit of the difference quotient as h approaches zero. This represents the slope of the secant line as it gets closer and closer to the tangent line. Watch as the secant line approaches the tangent line when h gets smaller.
There are several basic differentiation rules that make finding derivatives easier. The power rule states that the derivative of x to the n is n times x to the n minus one. Constants have a derivative of zero. The sum rule allows us to differentiate term by term. Here we see a cubic function and its derivative using these rules.
Differentiation has many practical applications. We can find maximum and minimum values by setting the derivative equal to zero. This is useful in optimization problems, physics for finding velocity and acceleration, and economics for marginal analysis. Here we see a parabola with its maximum point where the derivative equals zero.
To summarize what we have learned about differentiation: It is the process of finding instantaneous rates of change. The derivative represents the slope of the tangent line at any point. Basic differentiation rules make the process systematic. Applications include optimization, physics, and economics, making it an essential mathematical tool.