Partial Derivatives Example 1

In multivariable calculus you may be asked to find the partial derivatives. When deriving with respect to a variable, just treat all other variables as a constant.

Find all first partial derivatives of \(f(x,y) = {y^5} - 3xy\).

Let's start with the partial derivative with respect to x. See how there's only an x term in -3xy? Treat y as a constant.

\[\frac{{\partial f}}{{\partial x}} = - 3y\]

And now let's find the derivative with respect to y, so now treat x as a constant.

\[\frac{{\partial f}}{{\partial y}} = 5{y^4} - 3x\]

And you're done.

Please see our other examples on partial derivatives, including chain rule, product rule, quotient rule, and multivariable implicit differentiation:

Partial Derivatives Examples:

Partial Derivatives (Ex2)

Partial Derivatives (Ex3)

Partial Derivatives (Ex4)

Multivariable Chain Rule Examples:

Multivariable Chain Rule (Ex1)

Multivariable Chain Rule (Ex2)

Implicit Multivariable Differentiation:

Multivariable Implicit Differentiation

Return to the multivariable differentiation hub page:

Partial Derivatives, Multivariable Differentiation

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