It is provable in many ways by using other derivative rules. 1 2 3 Let where both f and g are differentiable and The quotient rule states that the derivative of h(x) is. We state this idea formally in a theorem. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Since rectangles that are "too big", as in (a), and rectangles that are "too little," as in (b), give areas greater/lesser than \(\displaystyle \int_1^4 f(x)\,dx\), it makes sense that there is a rectangle, whose top intersects \(f(x)\) somewhere on \(\), whose area is exactly that of the definite integral. Find absolute extrema on a closed region. For functions of several variables, find critical points using first partials and interpret them as relative extrema/saddle points using the second partials test. \): Differently sized rectangles give upper and lower bounds on \(\displaystyle \int_1^4 f(x)\,dx\) the last rectangle matches the area exactly.įinally, in (c) the height of the rectangle is such that the area of the rectangle is exactly that of \(\displaystyle \int_0^4 f(x)\,dx\). Use the chain rule for functions of several variables (including implicit differentiation). Please e-mail any correspondence to Duane Koubaīy clicking on the following address About this document. Your comments and suggestions are welcome.
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