Definition: PartialDerivative

Definition: Partial Derivative

Consider a function of several variables \(f(x,y)\). - The symbol \(\frac{\partial f}{\partial x}\) is read “the partial derivative of \(f\) with respect to \(x\)” or “the partial of \(f\) with respect to \(x\)”. - The symbol \(\frac{\partial f}{\partial y}\) is read “the partial derivative of \(f\) with respect to \(y\)” or “the partial of \(f\) with respect to \(y\)”. To calculate a partial derivative we treat all independent variables, except the one we are differentiating with respect to, as constants and use the derivative rules for a single variable function.

Let’s look at an example that includes the chain rule. Consider the function \(S(x,y) = 3(\pi x + 4y)^3\). - The partial of \(S\) with respect to \(x\) is \(\frac{\partial S}{\partial x} = 3(3(\pi x + 4y)^{2})(\pi)\). - And the partial of \(S\) with respect to \(y\) is \(\frac{\partial S}{\partial y} = 3(3(\pi x + 4y)^{2})(4)\).