BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5.4, Problem 12E
To determine

To find: use part 1 of the fundamental theorem of calculus to find the derivative of the function.

Expert Solution

Answer to Problem 12E

The derivative of the function is G(x)=-cos x

Explanation of Solution

Given information: The function is G(x)=-1xcost dt

Let’s remind of the fundamental theorem of calculus part 1:

The fundamental theorem of calculus part 1: If f is continuous on [a,b] then the function of g defined by

  g(x)=axf(t) dt where axb is continuous on [a,b] and differentiable on (a,b) and g(x)=f(x) .

First, we will use the properties of definite integral to match the form in fundamental theorem of calculus part 1.

So, x1cost dt=-1xcost dt where G(x)=-G(x)

Hence, G(x)=-1xcost dt

Now given, G(x)=-1xcost dt

Differentiation with respect to x,

  d G(x)dx=ddx[1xcost dt]

  G(x)=-cos x[where G(x)=F(x) ]

Thus, G(x)=-cos x

Hence, The derivative of the function is G(x)=-cos x

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