   Chapter 4.3, Problem 14E

Chapter
Section
Textbook Problem

# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h ( x ) = ∫ 1 x z 2 z 4 + 1 d z

To determine

To find:

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

Explanation

1) Concept:

2) The Fundamental Theorem of Calculus-Suppose f is continuous on [a, b]  then if  hs=auftdt, then h'=fu

3) 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)

4) In Leibnitz notation chain rule reads; if y=f(u) and u=g(x) are both differentiable functions then

dydx=dydu·dudx

2) Given:

hx=1xz2z4+1dz

3) Calculation:

Given that

hx=1xz2z4+1dz

Let u=x and substitute it in the above equation

Take derivative of both sides

h'x=ddx1uz2z4+1dz

Using chain rule

h'x=ddu1uz2z4+1dz·dudx

Here,

Use fundamental theorem of Calculus part 1) with fz=

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