   Chapter 4.3, Problem 16E

Chapter
Section
Textbook Problem

# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = ∫ 0 x 4 cos 2 θ d θ

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=0x4cos2θdθ

3) Calculation:

Here,

y=0x4cos2θdθ

Let u=x4 and substitute it in the above equation

Take derivative of both sides

y'=ddx0ucos2θdθ

Using chain rule

y'=ddu<

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