Application of Derivatives — Class 12 Mathematics

Derivatives measure rates of change and the local behaviour of functions, and this chapter applies them widely. The derivative as a rate of change links related quantities (for example, how a sphere's volume changes with its radius). A function is increasing where and decreasing where . The derivative also gives the slope of a tangent, so equations of tangents and normals follow at once. Maxima and minima are located using the first-derivative test and the second-derivative test: at a local maximum and , while at a local minimum and . These ideas drive optimisation problems — maximising area, minimising cost or material — that appear throughout applied mathematics.