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Partial Derivatives of an Integral Function

Multivariable calculus core

Let \(f(x, y) = \int_0^y (x^2 + tx) dt\). Find \(f_x\) and \(f_y\) and verify that \(f_{xy} = \f_{yx}\).
calculus-iiimultivariable-calculuspartial-derivativesmixed-partial-derivativesleibniz-rulefundamental-theorem-of-calculusdifferentiation-under-integral-signclairauts-theoremevaluate-partial-derivativeverify-equalitypolynomialsintegration

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What you need first

What this problem tests

Tests the ability to apply Leibniz's rule and the Fundamental Theorem of Calculus in the context of partial differentiation.

This problem is typical in a multivariable calculus course when introducing partial derivatives and Clairaut's theorem.

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