# Finding Partial Derivative of an Integral

• physman88
In summary, HallsofIvy suggests you start by thinking about ∫f = g … then ∂/∂x of (∫f between y and x).
physman88
Hey everyone. I am new here and i have a problem with some partials. We're studying partial derivatives in calculus III. I understand and all, but we haven't covered how to take a partial derivative of an integral. This problem showed up in my practice problems before our exam tomorrow.

The problem is as follows:

$$\frac{\partial}{\partial}x$$$$\int$$cos(t$$^{3}$$)dt

If you can't follow that.. then it says we need the first partials (x and y) of the integral of cos(t^3)dt.. (lower limit=y, and upper limit=x)

Any insight on how to start this problem?? Thanks for any help!

-Kev

physman88 said:
Hey everyone. I am new here and i have a problem with some partials. We're studying partial derivatives in calculus III. I understand and all, but we haven't covered how to take a partial derivative of an integral. This problem showed up in my practice problems before our exam tomorrow.

The problem is as follows:

$$\frac{\partial}{\partial}x$$$$\int$$cos(t$$^{3}$$)dt

If you can't follow that.. then it says we need the first partials (x and y) of the integral of cos(t^3)dt.. (lower limit=y, and upper limit=x)

Any insight on how to start this problem?? Thanks for any help!

-Kev
I couldn't follow it because you didn't put in the limits of integration, and that is crucial!

You should know from single variable calculus, the "Fundamental Theorem of Calculus":
$$\frac{d}{dt}\int_a^x f(t)dt= f(x)$$
where a is any constant.
From that it should be easy to find the partial derivative with respect to x.

To find the derivative with respect to y, remember that
$$\int_y^a f(t)dt= -\int_a^y f(t)dt$$

I understand that you couldn't follow it, but may I ask how to get the limits on the integral? Sorry for the mix-up.

HallsofIvy...your d/dt should be a d/dx for the fund. theor... the way you have it written it would be zero

physman88 said:
Any insight on how to start this problem??

Hey Kev!

Hint: start by thinking ∫f = g …

then ∂/∂x of (∫f between y and x)

= ∂/dx ([g] between y and x).

## 1. What is a partial derivative of an integral?

A partial derivative of an integral is a mathematical concept that involves finding the rate of change of a function with respect to one of its independent variables while holding the other variables constant. It is used in multivariable calculus to analyze how a function changes as one variable changes while the others remain fixed.

## 2. How is the partial derivative of an integral calculated?

The partial derivative of an integral is calculated using the fundamental theorem of calculus, which states that the derivative of an integral is equal to the integrand evaluated at the upper limit of integration. To find the partial derivative, we treat all other variables as constants and differentiate the function with respect to the desired variable.

## 3. What is the significance of finding partial derivatives of integrals?

Finding partial derivatives of integrals is important in many fields of science and engineering, such as physics, economics, and statistics. It allows us to understand how a function changes with respect to a specific variable, which is crucial in solving optimization problems and modeling real-world phenomena.

## 4. Can we find the partial derivative of any integral?

No, not all integrals have a closed-form solution and can be easily differentiated. Some integrals are too complex to be solved analytically, and in such cases, we use numerical methods to approximate the partial derivative.

## 5. How is the concept of partial derivatives of integrals related to total derivatives?

The concept of partial derivatives of integrals is closely related to total derivatives, as they both involve finding the rate of change of a function. However, partial derivatives focus on the change with respect to one specific variable, while total derivatives consider the overall change of the function with respect to all variables. Partial derivatives are a special case of total derivatives when all other variables are held constant.

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