Partial Derivatives Calculator

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Example

Created on 2024-06-20Asked by Isabella Gonzalez (Solvelet student)

Find the first-order partial derivatives

\frac{\partial f}{\partial x}

and

\frac{\partial f}{\partial y}

for the function

f(x, y) = x^2 + 3xy - 2y^2

Solution

To find the first-order partial derivatives

\frac{\partial f}{\partial x}

and

\frac{\partial f}{\partial y}

for the function

f(x, y) = x^2 + 3xy - 2y^2

: 1. Compute

\frac{\partial f}{\partial x}

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^2 + 3xy - 2y^2) = 2x + 3y.

2. Compute

\frac{\partial f}{\partial y}

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^2 + 3xy - 2y^2) = 3x - 4y.

Therefore, the first-order partial derivatives are

\frac{\partial f}{\partial x} = 2x + 3y

and

\frac{\partial f}{\partial y} = 3x - 4y

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mason Martinez on Solvelet

1. Compute the partial derivatives

\frac{{\partial f}}{{\partial x}}

and

\frac{{\partial f}}{{\partial y}}

for the function

f(x, y) = x^2y + y^3

.2. Determine the gradient vector of the function

g(x, y, z) = x^2yz + yz^2 + z^3

DefinitionPartial derivatives reveal the function’s rate of change with respect to one variable while other objectives are held constant. They are essential for studying multivariable calculus. Example: For f x,y = x2y + y^3, the expression derivative with respect to x is ∂x∂f=2xy.

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