Leibniz integral rule

[bsabrunosouza]

I'm solving a list of exercises, but this issue I couldn't solve

Statement: Find each partial derivative by treating, with the exception of one, the independent variables as constants and applying the rules of ordinary differentiation.

[MATH]f_{1}(x, y)\text{, where }f(x, y) = \int_{y}^{x} e^{-t^{3}} dt[/MATH]

 

[Deleted member 4993]

I'm solving a list of exercises, but this issue I couldn't solve

Statement: Find each partial derivative by treating, with the exception of one, the independent variables as constants and applying the rules of ordinary differentiation.

[MATH]f_{1}(x, y)\text{, where }f(x, y) = \int_{y}^{x} e^{-t^{3}} dt[/MATH]

Please show us what you have tried and exactly where you are stuck.

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[HallsofIvy]

First, "\(\displaystyle f_1\)" is the derivative of f with respect to the first listed variable which, here, is x, so \(\displaystyle f_1= \frac{\partial f}{\partial x}\).

This problem is asking you to find the derivative of \(\displaystyle f(x, y)= \int_y^x e^{-t^2}dt\) with respect to x, treating y as a constant.

"Liebniz' rule" says that \(\displaystyle \frac{d}{dx}\int_{\alpha(x)}^\beta(x) f(x, t)dt= f(x, \alpha(x))\frac{d\alpha}{dx}- f(x, \beta(x))\frac{d\beta}{dx}+ \int_{\alpha(x)}^{\beta(x)}\frac{\partial f}{\partial x}(x,t)dt\).

In this problem, the integrand, \(\displaystyle e^{-t^3}\), and lower limit of integration are not functions of x at all, while the upper limit of integration is just "x" so "Leibniz' rule" reduces to the "Fundamental Theorem of Calculus", \(\displaystyle \frac{d}{dx} \int_a^x f(t)dt= f(x)\)!