The Leibniz Rule: How to Differentiate Integrals with Respect to x

d/dx[(integral a-x) of f(t)]

To differentiate the integral with respect to x, we use the Leibniz rule, also known as the Leibniz integral rule or the differentiation under the integral sign

To differentiate the integral with respect to x, we use the Leibniz rule, also known as the Leibniz integral rule or the differentiation under the integral sign.

The Leibniz rule states that if we have a function of the form G(x) = ∫[a(x), b(x)] f(t) dt, where a(x) and b(x) are functions of x, then the derivative of G(x) with respect to x can be found using the following formula:

dG(x)/dx = f(b(x)) * db(x)/dx – f(a(x)) * da(x)/dx + ∫[a(x), b(x)] f'(t) dt

In our case, G(x) = ∫[a-x] f(t) dt, where a is a constant. So, a(x) = a and b(x) = a – x.

Let’s differentiate G(x) using the Leibniz rule:

dG(x)/dx = f(a – x) * d(a – x)/dx – f(a) * da/dx + ∫[a – x] f'(t) dt

Now, let’s simplify:

dG(x)/dx = -f(a – x) – f(a) * da/dx + ∫[a – x] f'(t) dt

So, the derivative of ∫[a – x] f(t) dt with respect to x is -f(a – x) – f(a) * da/dx + ∫[a – x] f'(t) dt.

Please note that this result assumes that f(t) is continuous and well-behaved within the given limits.

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