# Partial Derivative of a Definite Integral

• zl99
In summary, to find the partial derivatives of f(x,y) = ∫ (from -4 to x^3y^2) of cos(cos(t))dt, you should use the Fundamental Theorem of Calculus and the chain rule. First, calculate the anti-derivative of cos(cos(t)), then substitute the boundaries and differentiate it, taking into account the partial derivatives of x^3y^2. This will give you the partial derivatives with respect to x and y.
zl99
I'm trying to find the partial derivatives of:
f(x,y) = ∫ (from -4 to x^3y^2) of cos(cos(t))dt

and I am completely lost, any help would be appreciated, thanks.

Calculate the integral, i.e. the anti-derivative of ##cos^2(t)##, substitute the boundaries and differentiate it.

zl99 said:
I'm trying to find the partial derivatives of:
f(x,y) = ∫ (from -4 to x^3y^2) of cos(cos(t))dt

and I am completely lost, any help would be appreciated, thanks.

fresh_42 said:
Calculate the integral, i.e. the anti-derivative of ##cos^2(t)##, substitute the boundaries and differentiate it.
The integrand that zl99 wrote isn't ##\cos^2(t)## -- it's ##\cos(\cos(t))##.

I'd say that the strategy here is to use a form of the Fundamental Theorem of Calculus; i.e., that ##\frac d {dt} \int_a^x f(t)~dt = f(x)##. In this problem, I think you need to involve the chain rule. I haven't worked the problem, but that's the way I would go.

yes as @Mark44 said you must before use the FTC (Fundamental Theorem of Calculus) and multiply by the partial derivatives of ##x^3y^2## in one case you obtain the partial derivative respect ##x## and in the other case respect ##y## (you will use the chain rule for this)

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

A partial derivative of a definite integral is a mathematical concept that describes the rate of change of a function with respect to one of its variables while holding all other variables constant. It is used to find the sensitivity of the integral to changes in one of its variables.

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

The partial derivative of a definite integral is calculated by treating the integral as a function of the variable to be differentiated and using the rules of differentiation. The other variables are treated as constants and can be moved outside the integral.

## 3. What is the purpose of finding the partial derivative of a definite integral?

The purpose of finding the partial derivative of a definite integral is to analyze how the value of the integral changes with respect to one of its variables. This information is useful in many areas of science, including physics, economics, and engineering.

## 4. Can the partial derivative of a definite integral be negative?

Yes, the partial derivative of a definite integral can be negative. This indicates that as the value of the variable being differentiated increases, the value of the integral decreases. Similarly, a positive partial derivative indicates that the integral increases with increasing values of the variable.

## 5. Are there any applications of the partial derivative of a definite integral?

Yes, there are many applications of the partial derivative of a definite integral. Some examples include optimizing functions in economics, analyzing the motion of particles in physics, and determining the maximum power output in engineering problems.

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