An in-depth explanation of finding the derivative of an integral and the application of the chain rule

ⅆⅆx(∫x2sin(t4)ⅆt)=

To solve this problem, we can start by understanding the notation used and then applying the appropriate rules of differentiation

To solve this problem, we can start by understanding the notation used and then applying the appropriate rules of differentiation.

The expression ∫x^2sin(t^4)ⅆt represents the indefinite integral of x^2sin(t^4) with respect to t. This means that we need to find the antiderivative of the given function.

Let’s denote the antiderivative of x^2sin(t^4) with respect to t as F(t). Then, the expression becomes:

∫x^2sin(t^4)ⅆt = F(t) + C,

where C is the constant of integration.

Now, we need to differentiate the expression ∆∆x(F(t) + C) with respect to x. This involves applying the chain rule.

Using the chain rule, we have:

ⅆⅆx(F(t)) = ⅆⅆt(F(t)) * ⅆtⅆx,

where ⅆtⅆx represents the derivative of t with respect to x.

Since we want to find the derivative of the integral with respect to x, we can rewrite the above equation as:

ⅆⅆx(∫x^2sin(t^4)ⅆt) = ⅆⅆt(F(t)) * ⅆtⅆx.

Now, let’s find ⅆⅆt(F(t)). To do this, we need to compute the antiderivative of x^2sin(t^4) with respect to t.

Integrating the function x^2sin(t^4) with respect to t, we get:

F(t) = ∫x^2sin(t^4)ⅆt.

This integral is not easy to solve analytically, so we will leave it as F(t) for now.

Next, we need to find ⅆtⅆx, which represents the derivative of t with respect to x. This can be obtained by taking the reciprocal of ⅆxⅆt.

If we differentiate x with respect to t (ⅆxⅆt), we get:

ⅆxⅆt = 1.

To find ⅆtⅆx, we need to take the reciprocal, which gives us:

ⅆtⅆx = 1/ⅆxⅆt = 1/1 = 1.

Now, we have all the pieces we need to evaluate ⅆⅆx(∫x^2sin(t^4)ⅆt):

ⅆⅆx(∫x^2sin(t^4)ⅆt) = ⅆⅆt(F(t)) * ⅆtⅆx
= ⅆⅆt(F(t)) * 1
= ⅆⅆt(F(t)).

Therefore, the derivative of the given expression with respect to x is equal to ⅆⅆt(F(t)).

In summary, the derivative of ∫x^2sin(t^4)ⅆt with respect to x is ⅆⅆt(F(t)), where F(t) represents the antiderivative of x^2sin(t^4) with respect to t.

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