Understanding Derivatives
What Is a Derivative?
The derivative of a function measures how the output changes as the input changes. Formally, f'(x) = lim[h→0] (f(x+h) − f(x))/h. It gives the slope of the tangent line at any point on the curve.
The Power Rule
For a polynomial term axⁿ, the derivative is a·n·xⁿ⁻¹. This is the power rule — the most fundamental differentiation rule. It reduces the exponent by 1 and multiplies by the original exponent.
Geometric Interpretation
The derivative at a point is the slope of the tangent line to the curve at that point. A positive derivative means the function is increasing, negative means decreasing, and zero means it has a horizontal tangent (potential maximum or minimum).
Higher-Order Derivatives
The second derivative f''(x) measures the rate of change of the first derivative. It indicates concavity: positive means concave up (bowl-shaped), negative means concave down (dome-shaped). Inflection points occur where f''(x) = 0.
Applications
Derivatives are used in physics (velocity = derivative of position, acceleration = derivative of velocity), optimization (finding maxima and minima), economics (marginal cost, marginal revenue), biology (population growth rates), and machine learning (gradient descent).