find the derivative of:

g(x) = interal(a=7x, b=6x) (u+5)/(u-5) du

Hint:

interal(a=7x, b=6x) (u+5)/(u-5) du
= interal(a=0, b=6x) (u+5)/(u-5) du + interal(a=7x, b=0) (u+5)/(u-5) du

it's integral, not interal, sorry for the mistake

Using Leibnitz's integral rule,

d/dx ∫[7x,6x] (u+5)/(u-5) du
= 6*(6x+5)/(6x-5) - 7*(7x+5)/(7x-5)
= 60/(6x-5) - 70/(7x-5) - 1

Thank Steve. But how do you get 60 or 70?

or the -1?

To find the derivative of the given function g(x) = ∫[a=7x, b=6x] (u+5)/(u-5) du, we can use the Fundamental Theorem of Calculus.

First, let's break down the integral using the given hint:

∫[a=7x, b=6x] (u+5)/(u-5) du
= ∫[a=0, b=6x] (u+5)/(u-5) du + ∫[a=7x, b=0] (u+5)/(u-5) du

Next, we will use the Fundamental Theorem of Calculus, which states that if F(x) represents the integral of a function f(u) with respect to u, then the derivative of F(x) with respect to x is f(x).

Let's apply this rule to each portion of the integral:

For the first term, ∫[a=0, b=6x] (u+5)/(u-5) du:
Define a new function F(x) = ∫[a=0, b=x] (u+5)/(u-5) du.
Then, the derivative of F(x) with respect to x is f(x) = (x+5)/(x-5).

For the second term, ∫[a=7x, b=0] (u+5)/(u-5) du:
Define a new function G(x) = ∫[a=7x, b=0] (u+5)/(u-5) du.
Then, the derivative of G(x) with respect to x is g(x) = -(7x+5)/(7x-5).

Finally, to find the derivative of the original function g(x) = ∫[a=7x, b=6x] (u+5)/(u-5) du, we combine the derivatives of the two terms based on the given hint:

g(x) = f(x) + g(x)
= (x+5)/(x-5) - (7x+5)/(7x-5)

So, the derivative of g(x) is (x+5)/(x-5) - (7x+5)/(7x-5).