Find the derivative of the antiderivative

In summary, the conversation involves someone seeking help with a problem and mentioning trying a video tutorial. The FTOC and Chain Rule are mentioned as potential solutions. The person is also asking if the given rule can be used to compute F'(x).
  • #1
Umar
37
0
I know it seems pretty self explanatory, but I've tried to do this question and I've apparently gotten the wrong answer twice.

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If anyone can give me a clear solution to the problem, that would be greatly aooreciated. I initially tried to follow a video I saw online, but I think there is something different I need to do considering the bounds. Maybe splitting the integral?
 

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  • #2
By the FTOC and the Chain Rule, we have:

\(\displaystyle \frac{d}{dx}\int_{g(x)}^{h(x)} f(t)\,dt=f(h(x))\frac{dh}{dx}-f(g(x))\frac{dg}{dx}\)

Can you use this rule to compute $F'(x)$?
 

What is an antiderivative?

An antiderivative is the inverse operation of a derivative. It is a function that, when differentiated, gives the original function.

Why do we need to find the derivative of an antiderivative?

Finding the derivative of an antiderivative can help us solve more complex problems by breaking them down into simpler parts. It also allows us to find the rate of change of a function.

What are the steps to find the derivative of an antiderivative?

The steps to find the derivative of an antiderivative are:

  1. Identify the antiderivative function.
  2. Use the power rule or other derivative rules to differentiate the function.
  3. Check your answer using the reverse process of differentiation, integration.

What are some common functions and their corresponding antiderivatives?

Some common functions and their antiderivatives include:

  • Constant function: antiderivative is the original function multiplied by the variable.
  • Polynomial function: antiderivative is the original function with the exponent increased by one, divided by the new exponent.
  • Trigonometric function: antiderivative is the original function with the trigonometric identity added.

How is finding the derivative of an antiderivative related to the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus states that differentiation and integration are inverse operations of each other. Finding the derivative of an antiderivative is essentially using the Fundamental Theorem of Calculus in reverse, by first finding the antiderivative and then differentiating it to get the original function.

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