The Extension Leibnitz Rule, also known as the Leibnitz Integral Rule, is a mathematical formula used to find the derivative of an integral that depends on a parameter. It allows us to differentiate under the integral sign, which can be useful in solving complicated integration problems.
The formula for the Extension Leibnitz Rule is d/dx ∫ab f(x,t) dt = f(x,b) - f(x,a) + ∫ab ∂f(x,t)/∂x dt, where f(x,t) is the function being integrated and x is the variable of differentiation.
The Extension Leibnitz Rule is used when we need to find the derivative of an integral that contains a variable in both the integrand and the limits of integration. It is commonly used in physics and engineering to solve problems involving continuous systems.
The Extension Leibnitz Rule assumes that the function being integrated is continuous and that its derivative with respect to the variable of differentiation exists and is also continuous. Additionally, the limits of integration must not depend on the variable of differentiation.
One example is using the rule to find the derivative of the function ∫ax sin(t)/t dt. Another example is using the rule to solve differential equations, such as in the case of finding the derivative of the function ∫ax etx f(t) dt to solve the differential equation f'(t) = tf(t).