One can ask whether the vehicles are getting closer or further apart and at what rate at the moment when the northbound vehicle is 3 miles north of the intersection and the westbound vehicle is 4 miles east of the intersection.īig idea: use chain rule to compute rate of change of distance between two vehicles. Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner. Chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. The chain rule can be used to find whether they are getting closer or further apart.įor example, one can consider the kinematics problem where one vehicle is heading west toward an intersection at 80mph while another is heading north away from the intersection at 60mph. We use the chain rule when differentiating a function of a function, like f(g(x)) in. The next two examples illustrate 'functional' and 'Leibniz' methods of attacking the same problem using the chain rule.The chain rule is a method to compute the derivative of the functional composition of two or more functions. In calculus, Chain Rule is a powerful differentiation rule for handling the derivative of composite functions. These are two really useful rules for differentiating functions. However, a fully rigorous proof is beyond the secondary school level. We first explain what is meant by this term and then learn about the Chain Rule which is the technique used to perform the differentiation. The proof above is not entirely rigorous: for instance, if there are values of \(\Delta x\) close to zero such that \(g(x \Delta x) - g(x) = 0\), then we have division by zero in the first limit.
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