Fundamental Theorem of Calculus to find Derivative

Then,g'(x) = u'v + uv' where u' = (1/2)x^(-1/2)cos(x) + (1/2)\sqrt x sin(x)and,v' = \frac{ln(x)cos(x)}{x} + \frac{sin(x)}{x} - \frac{ln(a)cos(a)}{a} - \frac{sin(a)}{a}Therefore,In summary, using the Fundamental Theorem of Calculus, the derivative of the function g(x) is g'(x) = u'v + uv', where u is defined as (1/2)x^(-1/2)cos(x) + (1/
  • #1
jeeves_17
10
0
Use the Fundamental Theorem of Calculus to find the derivative of the function



g(x) = [tex]\sqrt{x}\int sinx[/tex] Ln(t) [tex]\frac{cos(t)}{t}[/tex] dt



g'(x) = lnx cosx / x. By integrating this function, you receive the function g(x). Then by differentiating g(x) you receive g'(x) which is what is given, according to FTCI.


I was told I got this completely wrong. (out of 5) So looking for some help. Thanks in advance.



 
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  • #2
I do not understand something. Could you possibly rewrite the original problem?
Is it:

[tex]g(x)=\sqrt{x}\int{sin(x)dx} *\int{\frac{ln(t)*cos(t)}{t}dt} [/tex]


Regards.
 
  • #3
As stated, the function [tex]g(x)[/tex] doesn't look quite right. Is it supposed to be defined such that,

[tex]g(x) = \sqrt x sin(x) \int_{a}^x \frac{ln(t)cos(t)}{t} \, dt[/tex]

If so, then use the product rule of differentiation because [tex]g(x)[/tex] can be defined in terms of the product of two functions [tex]u[/tex] and [tex]v[/tex] where,

[tex]u = \sqrt x sin(x)[/tex]

[tex]v = \int_{a}^x \frac{ln(t)cos(t)}{t} \, dt[/tex]
 

Related to Fundamental Theorem of Calculus to find Derivative

1. What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a fundamental concept in calculus that relates the two main operations of calculus: integration and differentiation. It states that the derivative of a function can be calculated by evaluating the integral of that function's derivative.

2. How do you use the Fundamental Theorem of Calculus to find the derivative of a function?

To find the derivative of a function using the Fundamental Theorem of Calculus, you first need to evaluate the integral of the function's derivative. This integral will give you the original function plus a constant. Then, simply take the derivative of the original function to eliminate the constant and find the actual derivative.

3. What is the difference between the First and Second Fundamental Theorem of Calculus?

The First Fundamental Theorem of Calculus states that the derivative of a function can be found by evaluating the integral of that function's derivative. The Second Fundamental Theorem of Calculus, on the other hand, states that the integral of a function can be evaluated by using the function's antiderivative.

4. Why is the Fundamental Theorem of Calculus important in calculus?

The Fundamental Theorem of Calculus is important in calculus because it allows for the simplification of complex functions by relating the two main operations of calculus. It also allows for the calculation of definite integrals, which have many practical applications in various fields of science and engineering.

5. Can the Fundamental Theorem of Calculus be applied to all functions?

Yes, the Fundamental Theorem of Calculus can be applied to all continuous functions. However, certain conditions must be met for the theorem to be applied, such as the function being continuous on the interval of integration and having a continuous derivative. If these conditions are not met, then the theorem may not be applicable.

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