I heard that when we are proving Leibniz's formula for differentiating an integral, we use chain rule i.e.

Question

I heard that when we are proving Leibniz's formula for differentiating an integral, we use chain rule i.e.

dw/dx=(�Ýw/�Ýu)(du/dx) + =(�Ýw/�Ýv)(dv/dx) + (�Ýw/�Ýx)

where u is the upper limit and v is the lower limit of integration, and the w is integral of f(x,y) with respect to y.

how is this chain rule work?

i tried to c how it works, but i cannot get it.

Answer 1

To figure it out graph a function from x = a to x = b

Then look for example at how the area under the function changes at x = b as b changes
It is the value of the function at b times the change of b

Answer 2

The chain rule is a fundamental rule in calculus that allows us to find derivatives of composite functions. In this case, the chain rule is used to differentiate an integral that depends on a variable inside the limits of integration. Let me explain how it works step by step:

1. Start with the integral function w = ∫[a to b] f(x, y) dy, where f(x, y) is the integrand and y is the integration variable.

2. We want to differentiate w with respect to x, so we calculate dw/dx.

3. To differentiate w with respect to x, we need to consider how the limits of integration, a and b, change with respect to x. Therefore, we need to use the chain rule.

4. Consider the first term of the chain rule: (d⁡w/du) × (du/dx). This term accounts for the change in the upper limit of integration, u, with respect to x. Since u depends on x, we need to differentiate it with respect to x to find du/dx.

5. Similarly, consider the second term of the chain rule: (d⁡w/dv) × (dv/dx). This term accounts for the change in the lower limit of integration, v, with respect to x. We differentiate v with respect to x to find dv/dx.

6. Finally, consider the third term: (d⁡w/dx). This term accounts for how the integrand f(x, y) itself depends on x. We differentiate f(x, y) with respect to x to find (d⁡f/⁡dx), and then multiply it by 1 since dx/dx = 1.

7. Summing up these three terms gives us the derivative dw/dx.

So, in summary, the chain rule is applied to differentiate the integral function by considering the changes in the limits of integration and the integrand with respect to the differentiating variable.