Differentiation is one of the fundamental concepts in calculus. It allows us to find the instantaneous rate of change of a function at any point. Think of it like a car's speedometer - while your position changes over time, the speedometer tells you exactly how fast you're moving at each moment. Mathematically, we write the derivative as f prime of x, or dy dx. Geometrically, the derivative represents the slope of the tangent line to the curve at any given point.
To understand how derivatives work, we need to see the connection between secant lines and tangent lines. A secant line connects two points on a curve. As we move the second point closer and closer to the first point, something remarkable happens - the secant line approaches the tangent line. This limiting process gives us the formal definition of a derivative: f prime of x equals the limit as h approaches zero of f of x plus h minus f of x, all divided by h. For example, with f of x equals x squared, this limit equals 2x.
Now let's learn the fundamental rules of differentiation. The power rule states that the derivative of x to the n is n times x to the n minus 1. The constant rule tells us that the derivative of any constant is zero. For constant multiples, we can factor out the constant. The sum and difference rules allow us to differentiate term by term. Let's apply these rules to an example: f of x equals 3x squared plus 5x minus 2. Using our rules, f prime of x equals 6x plus 5.
For more complex functions, we need the product and quotient rules. The product rule states that the derivative of f times g equals f prime times g plus f times g prime. A helpful memory device is: first times derivative of second, plus second times derivative of first. The quotient rule for f over g equals f prime g minus f g prime, all over g squared. Let's see an example: the derivative of x squared sine x equals 2x sine x plus x squared cosine x, using the product rule.
The chain rule is essential for differentiating composite functions - functions within functions. The rule states that the derivative of f of g of x equals f prime of g of x times g prime of x. Think of it as the outside-inside method: take the derivative of the outside function, then multiply by the derivative of the inside function. For example, the derivative of x squared plus 1 to the third power is 3 times x squared plus 1 squared, times 2x. This completes our fundamental differentiation toolkit.