# What is the derivative of # int_x^(x^2) t^3 \ dt # wrt #x#?

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# d/dx \ int_x^(x^2) t^3 \ dt =2x^7 - x^3 #

If asked to find the derivative of an integral then you should not evaluate the integral but instead use the fundamental theorem of Calculus.

The FTOC tells us that:

(ie the derivative of an integral gives us the original function back).

We are asked to find:

And so:

And therefore differentiating we get:

(notice the upper bounds of the first integral are not in the correct format for the FTOC to be applied, directly). We can manipulate the definite integral using a substitution and the chain rule. Let:

The substituting into the first integral we get:

And now the derivative of the integral is in the correct form for the FTOC to be applied, giving:

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Another way to do this is to apply Leibniz's integral rule:

In this case, we have:

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The derivative of ( \int_{x}^{x^2} t^3 , dt ) with respect to ( x ) is ( 2x^5 - x^3 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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