Differentiation is a fundamental operation in calculus that finds the instantaneous rate of change of a function. It measures how a function's output changes as its input changes. Here we see a parabola, and the red line shows the tangent at different points, representing the instantaneous rate of change.
The result of differentiation is called the derivative of the function. The derivative at a specific point represents the slope of the tangent line to the function's graph at that point. For example, at x equals 2, the derivative of x squared is 4, which is the slope of the red tangent line.
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 1. We also have rules for constants, sums, and products. The derivative is formally defined using limits, as the secant line approaches the tangent line when h approaches zero.
Let's work through a practical example. To find the derivative of 3x cubed plus 2x squared minus 5x plus 1, we apply the power rule to each term. The derivative is 9x squared plus 4x minus 5. At x equals 1, the derivative value is 8, which represents the slope of the tangent line at that point.
To summarize what we've learned about differentiation: It finds the instantaneous rate of change of functions. The derivative represents the slope of the tangent line at any point. We use basic rules like the power rule to calculate derivatives efficiently. Derivatives are formally defined using limits, and they have wide applications in physics, economics, and optimization problems.