   Chapter 4.3, Problem 17E

Chapter
Section
Textbook Problem

# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = ∫ x π / 4 θ tan 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=xπ4θ tan(θ)dθ

3) Calculation:

Given that

y=xπ4θtan(θ)dθ

But

xπ4θtan(θ)dθ=-π4xθtan(θ)dθ

Substitute it in the above equation, so

y=-π4xθtan(θ)dθ

Let u=x and substitute it in the  above equation and take derivative of both sides

y'=ddx-</

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