Leibnitz's Rule Statement And It's Proof
WELCOME TO MATHEMATICS I n this mathematics session I shall prove that , under suitable conditions, ' the derivative of the integral and the integral of the derivative are equal ' , and consequently , ' the two repeated integrals are equal for continuous functions '. Leibnitz's Rule In Mathematics: If f is defined and continuous on the rectangle R = [a,b;c,d] , and if (i) fₓ(x,y) exists and is continuous on the rectangle R , and d (ii) g(x) = ∫ f(x,y) dy , for x∈ [a,b] c then g is differentiable on [a,b] and d g'(x) = ∫ fₓ(x,y) dy c d d i.e., d/dx {∫ f(x,y) dy }=∫ ∂f(x,y)/∂x dy c c Proof Of Leibnitz's Rule In Mathematics : Since fₓ (∂f/∂x) exists on R ,