Labware - MA35 Multivariable Calculus - Single Variable Calculus
 MA35 Labs 1 » Single Variable Calculus Contents1.2 Differentiation 1.2.1 Derivatives 1.2.2 Critical Points 1.2.4 Chain Rule 1.2.5 Differentiability 1.4 Integration Search

Chain Rule

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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).

Demos

 Chain Rule Here we see the rate of change x '(t) of the function x(t) with respect to t represented by the magnitude of the vector drawn along the x-axis. To find the rate of change of f(x(t)) with respect to t, we use x '(t) as the horizontal component of the tangent vector to the parametrized curve at (x(t), f(x(t))), and our result is f '(x(t))x '(t).

 Chain rule A common misconception leads to an incorrect calculation of the derivative of a composition of functions. This demo contrasts the resulting error with the correct answer, given by the chain rule. The first window graphs two functions: f(x) (in gray) and its derivative, f '(x) (in green). In the second window, we see three curves, a graph of the composition f(g(x)) (in gray), an incorrect calculation of the derivative of this composed function with respect to x (in green), and a correct calculation (in red). The incorrect calculation, common among those unfamiliar with the chain rule, is given by f '(g(x)). (For example, one might be inclined to think that the derivative with respect to x of sin(2x) is cos(2x)). One useful way to think about the chain rule is to consider the rate at which we are "moving through" the values of the function f(x). For example, though sin(x) and sin(2x) both express rates of change as one moves through the values of the sine function, the function sin(2x) goes through those values twice as fast as does the function sin(x), which makes the graph twice as steep, and requires us to correct our mistake (cos(2x)) by multiplying by 2. The "correction factor" of g' (x) tells us how fast we are moving through the values of the function f(x). The correct graph of the derivative of f(g(x)) with respect to x is shown in red.

Exercises

• 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.