Applying Leibniz's Rule for Differentiating an Integral

In summary, differentiating an integral is used to find the rate of change of a function at a particular point. It is the inverse operation of integration and is linked by the Fundamental Theorem of Calculus. Any integral can be differentiated as long as the function is continuous and differentiable on the interval of integration. To differentiate an integral, various differentiation rules can be used depending on the form of the integral. Real-world applications of differentiating an integral include finding velocity, determining population growth, calculating slope, and analyzing physical systems in fields such as economics, engineering, and physics.
  • #1
rootX
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Homework Statement



[tex]\int_{x}^{2\,{x}^{2}+1}sin{t}^{2}dt[/tex]

I need to take differential of that

Homework Equations



Fundamental theorem of calculus

The Attempt at a Solution



I know 't' is a dummy var, so I replace it with x,

and then
get
sin((2x^2+1)^2)-sin(x^2)
as answer. But I am not very sure about my answer.

Can anyone please check my solution?

Thanks.
 
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  • #2
Look up Leibniz's Rule. After that it's just a plug and chug:

Edit: Leibniz Rule, not Theorem
 
Last edited:

1. What is the purpose of differentiating an integral?

The purpose of differentiating an integral is to find the rate of change of a function at a particular point. This is useful in many applications, such as determining the velocity of an object or the growth rate of a population.

2. How is differentiation related to integration?

Differentiation and integration are inverse operations. This means that differentiating an integral will bring you back to the original function, and integrating a derivative will also bring you back to the original function. They are also linked by the Fundamental Theorem of Calculus.

3. Can any integral be differentiated?

Yes, any integral can be differentiated as long as the function is continuous and differentiable on the interval of integration. If the function is discontinuous or not differentiable, the derivative may not exist or may be undefined at certain points.

4. What is the process for differentiating an integral?

To differentiate an integral, you can use the power rule, product rule, quotient rule, chain rule, or other differentiation rules depending on the form of the integral. The integral should be rewritten in terms of x and then differentiated using these rules.

5. What are some real-world applications of differentiating an integral?

Differentiating an integral has many real-world applications, including finding the velocity of an object, determining the rate of change of a population, calculating the slope of a curve, and analyzing the behavior of physical systems. It is also used in fields such as economics, engineering, and physics to model and understand complex systems.

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