### Partial Derivatives Example 2

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 \(f(x,t) = {e^{ - t}}\cos (\pi x)\)

We'll find the partial derivative with respect to x first, so treat everything else as a constant.

\[\frac{{\partial f}}{{\partial x}} = - \pi {e^{ - t}}\sin (\pi x)\]

Now find the derivative with respect to y, so you treat x as a constant.

\[\frac{{\partial f}}{{\partial y}} = - {e^{ - y}}\cos (\pi x)\]

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

Partial Derivatives Examples:

Partial Derivatives (Ex1)

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