   Chapter 4.3, Problem 18E

Chapter
Section
Textbook Problem

# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = ∫ sin x 1 1 + t 2 d t

To determine

To find:

The derivative of the function using part 1) of the Fundamental Theorem of calculus

Explanation

1) Concept:

i. The Fundamental Theorem of Calculus-Suppose f is continuous on [a, b]  then,

if hs=auftdt, then h'=fu

ii. Chain rule- Let Fx=fgx, if g is differentiable at x and f is differentiable at g(x) then F'x=f'gx·g'(x)

iii. In Leibnitz notation if y=f(u) and u=g(x) are both differentiable functions then

dydx=dydu·dudx

2) Given:

y=sinx11+t2dt

3) Calculation:

Here,

y=sinx11+t2dt

We may use property of integral to write

y=sinx11+t2dt=-1sinx1+t2dt

Substitute it in the above equation

y=-1sinx1+t2dt

Let u=sin(x) and substitute it in the  above equation

Take derivative of both sides

y'=ddx-1sinx1+t2dt

Using chain rule

y'

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