04 The Chain Rule and Extended Power Rule for Composite Functions
In practical scientific and engineering formulations, functions are frequently nested as composites: y = f(g(x)). The Chain Rule dictates that the derivative of the composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Extended Power Rule: If y = [u(x)]n, where u(x) is a differentiable function of x, then:
dy/dx = n × [u(x)]n - 1 × u'(x).
Differentiating Radical Functions: For square roots y = √u(x), applying the chain rule directly yields a rapid computational shortcut: dy/dx = u'(x) / [2 √u(x)]. That is, the derivative of a square root equals the derivative of what is inside the root divided by twice the root itself.
Implicit Differentiation: For relations where y cannot be explicitly isolated as y = f(x) (such as the equation of a circle x² + y² = r² or x³ + y³ = 3axy): we differentiate both sides with respect to x, remembering by the chain rule that d/dx [yn] = n y^(n-1) (dy/dx), and then algebraically solve for the term dy/dx.