The derivative of a function measures how fast the function is changing at any given point. Geometrically, it represents the slope of the tangent line to the curve at that point. This concept is fundamental in calculus and has applications throughout mathematics, physics, and engineering.
The derivative is formally defined using limits. We take the slope of a secant line between two points on the curve, then let the distance h approach zero. As h gets smaller, the secant line approaches the tangent line, and its slope approaches the derivative.
Instead of using the limit definition every time, we use differentiation rules. 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. The sum rule allows us to differentiate term by term. For example, to find the derivative of 3x squared plus 2x plus 1, we apply these rules step by step to get 6x plus 2.
The chain rule is essential for differentiating composite functions. When we have a function inside another function, we multiply the derivative of the outer function by the derivative of the inner function. For example, to find the derivative of (2x + 1) cubed, we treat it as f of g of x, where f(u) equals u cubed and g(x) equals 2x + 1. The result is 6 times (2x + 1) squared.
Derivatives have numerous practical applications. They help us find rates of change, solve optimization problems, analyze motion, and sketch curves. For example, we can find maximum and minimum points where the derivative equals zero. To find any derivative, identify the function type, apply the appropriate differentiation rules, and simplify the result. With practice, differentiation becomes a powerful tool for solving real-world problems.