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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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

The derivative of the function h(x)=1xz2z4+1dz using Part 1 of the fundamental theorem of calculus.

Explanation

Given information:

Show the integral function as follows:

The integral function is h(x)=1xz2z4+1dz (1)

The Part 1 of the fundamental theorem of calculus is shown below:

F(x)=axf(t)dt

Consider that part 1 of the fundamental theorem is valid under following conditions:

  • If f is a continuous function on the interval [a,b].
  • The function is differentiable on the interval, and F(x)=f(x).

Calculation:

Consider u=x (2)

Differentiate Equation (2) with respect to x.

dudx=12x (3)

Find the value of dhdx using the relation shown below.

dhdx=dhdududx (4)

Substitute h(x) in Equation (4).

dhdx=ddu1xz2z4+1dz×dudx (5)

Substitute u for x in Equation (5)

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