Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule expresses the derivative of their composition f ∘ g — the function which maps x to f ( g ( x ) ) {displaystyle f(g(x))} — in terms of the derivatives of f and g and the product of functions as follows: In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule expresses the derivative of their composition f ∘ g — the function which maps x to f ( g ( x ) ) {displaystyle f(g(x))} — in terms of the derivatives of f and g and the product of functions as follows: Alternatively, by letting F = f ∘ g (equiv., F(x) = f(g(x)) for all x), one can also write the chain rule in Lagrange's notation, as follows: The chain rule may also be rewritten in Leibniz's notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x (i.e., y and z are dependent variables), then z, via the intermediate variable of y, depends on x as well. In which case, the chain rule states that: The two versions of the chain rule are related, in the sense that if z = f ( y ) {displaystyle z=f(y)!} and y = g ( x ) {displaystyle y=g(x)!} , then In integration, the counterpart to the chain rule is the substitution rule. The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative of a + b z + c z 2 {displaystyle {sqrt {a+bz+cz^{2}}}} as the composite of the square root function and the function a + b z + c z 2 {displaystyle a+bz+cz^{2}!} . He first mentioned it in a 1676 memoir (with a sign error in the calculation). The common notation of chain rule is due to Leibniz. Guillaume de l'Hôpital used the chain rule implicitly in his Analyse des infiniment petits. The chain rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery. Suppose that a skydiver jumps from an aircraft. Assume that t seconds after his jump, his height above sea level in meters is given by g(t) = 4000 − 4.9t2. One model for the atmospheric pressure at a height h is f(h) = 101325 e−0.0001h. These two equations can be differentiated and combined in various ways to produce the following data: Here, the chain rule gives a method for computing (f ∘ g)′(t) in terms of f′ and g′. While it is always possible to directly apply the definition of the derivative to compute the derivative of a composite function, this is usually very difficult. The utility of the chain rule is that it turns a complicated derivative into several easy derivatives.