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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

h ( x ) = 2 1 / x sin 4 t 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. Chain rule in Leibnitz notation: if y=f(u) and u=g(x) are both differentiable functions then

dydx=dydu·dudx

2) Given:

hx=21xsin4(t)dt

3) Calculation:

Given that

hx=21xsin4(t)dt

Let u=1x and substitute it in the above equation

Take derivative of both sides

h'x=ddx2usin4(t)dt

Using chain rule

h'x=ddu

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