Derivative of a integral function?

[pellman]

How does one work the following?

$$\frac{d}{dx}\int^x_y f(x,u)du$$

I know that (given certain assumptions about the function f)

$$\frac{d}{dx}\int^x_y f(w,u)du=f(w,x)$$

and

$$\frac{d}{dx}\int^c_y f(x,u)du=\int^c_y \frac{df}{dx}(x,u)du$$

but how do we put them together?

 

[Pere Callahan]

$$\frac{d}{dx}\int^x_y f(x,u)du = \int^x_y \frac{d}{dx}f(x,u)du+f(x,x).$$

 

[pellman]

Thanks, Pere.

 

[HallsofIvy]

In general, Lagrange's formula:
$$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x,t)dt= f(x,\beta(x))\frac{d\beta}{dx}- f(x, \alpha(x))\frac{d\alpha}{dx}+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f}{\partial x} dt$$

In this particular problem y is independent of both x and u and can be treated as a constant: dy/dx= 0.

 

[CellCoree]

The derivative of an integral function is a fundamental concept in calculus. It is used to determine the rate of change of a function with respect to its independent variable. In order to work with the given expression, we can use the fundamental theorem of calculus, which states that the derivative of an integral function is equal to the integrand evaluated at the upper limit of integration. In this case, the upper limit is x and the lower limit is y. Therefore, we can rewrite the expression as follows:

\frac{d}{dx}\int^x_y f(x,u)du = f(x,x)

where f(x,x) represents the integrand evaluated at x. This simplifies the expression and allows us to easily calculate the derivative.

In cases where the integrand also depends on the independent variable x, we can use the chain rule to find the derivative. This results in the following expression:

\frac{d}{dx}\int^x_y f(x,u)du = f(x,x) + \int^x_y \frac{\partial f}{\partial x}(x,u)du

where \frac{\partial f}{\partial x}(x,u) represents the partial derivative of the integrand with respect to x. This allows us to handle more complex functions and still find the derivative of the integral function.

In summary, to find the derivative of an integral function, we use the fundamental theorem of calculus and the chain rule to simplify the expression and evaluate the integrand at the upper limit of integration. With these tools, we can work with a wide range of functions and determine their rate of change with respect to their independent variable.