![]() Some content on this page may previously have appeared on Citizendium. Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ( multivariate ), rather than just one. Then is differentiable at with derivative Thus D F is a linear map from and D G is a linear map from. Now let and be functions with F having derivative D F at and G having derivative D G at. Chain Rule: Since the composition of linear mappings expressed as matrices are matrix multiplication, the Chain Rule takes the alternative form. w x2 z y4 x t3 +7, y cos(2t), z 4t w x 2. Given the following information use the Chain Rule to determine dw dt d w d t. The extension of the chain rule to multivariable functions may be achieved by considering the derivative as a linear approximation to a differentiable function. Given the following information use the Chain Rule to determine dz dt d z d t. Which is easy to remember, because it as if d y in the numerator and the denominator of the right hand side cancels. In mnemonic form the latter expression is ![]() In order to convert this to the traditional ( Leibniz) notation, we notice So the rate at which z varies in terms of x is the product, and substituting we have the chain rule The rate at which z varies in terms of y is given by the derivative, and the rate at which y varies in terms of x is given by the derivative. Suppose that y is given as a function and that z is given as a function. In calculus, the chain rule describes the derivative of a "function of a function": the composition of two function, where the output z is a given function of an intermediate variable y which is in turn a given function of the input variable x.
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