"Differentiating under the integral" is a mathematical technique that allows us to find the derivative of a function that is defined by an integral. It involves differentiating both the limits of integration and the integrand with respect to a variable, and then integrating the resulting expression.
"Differentiating under the integral" is useful in situations where the function being integrated is dependent on a parameter, and we need to find the derivative of the integral with respect to that parameter. It is also useful when the integrand is too complex to integrate directly.
The steps to "Differentiating under the integral" are as follows:1. Identify the variable of differentiation and the limits of integration2. Differentiate the limits of integration with respect to the variable3. Differentiate the integrand with respect to the variable4. Plug in the differentiated limits and integrand into the integral5. Integrate the resulting expression to find the derivative
Yes, there are two special cases when using "Differentiating under the integral":1. If the integrand is independent of the variable of differentiation, then the derivative of the integral is simply the integrand multiplied by the derivative of the limits.2. If the limits of integration are constants, then the derivative of the integral is equal to 0.
"Differentiating under the integral" has various applications in physics, engineering, and economics. It can be used to find the rate of change of a quantity, the slope of a curve, or the optimization of a system. Some examples include calculating the velocity of a moving object, finding the maximum profit in a business model, and solving differential equations.