Labware - MA35 Multivariable Calculus - Single Variable Calculus



Chain Rule


The chain rule expresses the derivative of the composition of two functions in terms of the derivatives of the functions. Specifically, if x(t) is a differentiable function of t and f(x) is a differentiable function of x, then y(t) = f(x(t)) is a differentiable function of t and y'(t) = f'(x(t)) x'(t).

If we have a curve (x(t), y(t)) in the plane, then the velocity vector is given by (x'(t),y'(t)) and the slope of the velocity vector, when x'(t) ≠ 0, is given by y'(t)/x'(t). If y(t) = f(x(t)), then the slope of the tangent line to the graph y = f(x) is f'(x), so the slope at x(t) is f'(x(t)). It follows that f'(x(t)) = y'(t)/x'(t) and the chain rule follows once we multiply by x'(t).



  • 1. In the second demo, try g(x) = 10 * x and then try g(x) = 0.1 * x with a few different expressions for f(x). Describe qualitatively what is happening.
  • 2. Describe an intuitive way to think about the chain rule for the special case g(x) = -x.