Differentiation from first principles is a fundamental concept in calculus. It finds the derivative of a function using the limit definition. The derivative represents the instantaneous rate of change at any point on a curve. We start with a function and examine how the slope of a secant line changes as we make the interval smaller and smaller.
The process involves three key steps. First, we choose two points on the curve: one at x and another at x plus h. Second, we calculate the slope of the secant line connecting these points using the formula f of x plus h minus f of x, all divided by h. Finally, we apply the limit as h approaches zero to find the instantaneous rate of change, which gives us the derivative.
Let's work through a concrete example. We'll find the derivative of f of x equals x squared using first principles. We start by expanding f of x plus h, which gives us x squared plus 2 x h plus h squared. Then we substitute into our difference quotient formula and simplify to get 2 x plus h. Finally, taking the limit as h approaches zero gives us the derivative: 2 x.
Let's try another example with a linear function: f of x equals 3 x plus 2. When we apply first principles, we find that f of x plus h equals 3 x plus 3 h plus 2. The difference quotient simplifies to 3 h over h, which equals 3. Since this doesn't depend on h, the limit is simply 3. This confirms that the derivative of a linear function is just its slope.
To summarize what we've learned about differentiation from first principles: This fundamental method uses the limit definition to find derivatives by examining how secant line slopes behave as the interval approaches zero. It provides the instantaneous rate of change and forms the theoretical foundation for all differentiation rules we use in calculus.