gives a uniform (independent of ) bound on . The difference quotients converge pointwise to the partial derivative fx by the assumption that the partial derivative exists. The above argument shows that for every sequence {δn} → 0, the sequence is uniformly bounded and converges pointwise to fx. The bounded convergence theorem states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. In particular, the limit and integral may be exchanged for every sequence {δn} → 0. Therefore, the limit as δ → 0 may be passed through the integral sign. Variable limits form For a continuous real valued function g of one real variable, and real valued differentiable functions and of one real variable, This follows from the chain rule and the First Fundamental Theorem of Calculus. Define and (The lower limit just has to be some number in the domain of ) Then, can be written as a composition: . The Chain Rule then implies that By the First Fundamental Theorem of Calculus, . Therefore, substituting this result above, we get the desired equation: Note: This form can be particularly useful if the expression to be differentiated is of the form: Because does not depend on the limits of integration, it may be moved out from under the integral sign, and the above form may be used with the Product rule, i.e., General form with variable limits Set where a and b are functions of α that exhibit increments Δa and Δb, respectively, when α is increased by Δα. Then, A form of the mean value theorem, , where a < ξ < b, may be applied to the first and last integrals of the formula for Δφ above, resulting in Divide by Δα and let Δα → 0. Notice ξ1 → a and ξ2 → b. We may pass the limit through the integral sign: again by the bounded convergence theorem. This yields the general form of the Leibniz integral rule, Alternative proof of the general form with variable limits, using the chain rule The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the basic form of Leibniz's Integral Rule, the multivariable chain rule, and the First Fundamental Theorem of Calculus. Suppose is defined in a rectangle in the plane,