Question

DOUBLE POINTS
Find the derivative.

Answer 1

`4√(16x^2+1)`

Explanation:

Given
`(d)/(dx) \int\limits^(4x)_ {1} \, √(t^2+1)\ dt`

Using Fundamental Theorem of Calculus

`(d)/(dx) \int\limits^(x)_ {a} \, f(t)dt =f(x)\ \ \ for\ any\ constant\ a`

`(d)/(dx) \int\limits^(4x)_ {1} \, √(t^2+1)\ dt=(d(4x))/(dx)(d)/(dx)\int\limits^(4x)}_(1)\, √(t^2+1)dt`

Now,
`(d(4x))/(dx)=4\ and\ (d)/(d(4x)) \int\limits^(4x)_ {1} \, √(t^2+1)\ dt=√((4x^2)+1)`

Hence plugging these results we get:

`(d)/(dx) \int\limits^(4x)_ {1} \, √(t^2+1)\ dt=4√((4x)^2+1)\\\\=4√(16x^2+1)`