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2.5 Differentiating using calculus rules

Looking more closely at the calculations on the previous frame, we can make out the following general rule: To determine the partial derivative ∂⁡f∂⁡x\frac{\partial f}{\partial x}, simply differentiate with respect to xx, while thinking of the variable yy as ‘constant’ for the purposes of the calculation.

Similarly, to determine ∂⁡f∂⁡y\frac{\partial f}{\partial y}, differentiate with respect to yy while keeping xx constant.

Example.

Find ∂⁡f∂⁡x\frac{\partial f}{\partial x} and ∂⁡f∂⁡y\frac{\partial f}{\partial y} for f=x3+x2⁢y+y2f=x^{3}+x^{2}y+y^{2}.

We have

∂⁡f∂⁡x= 3⁢x2+2⁢x⁢y, while⁢∂⁡f∂⁡y=x2+2⁢y.\frac{\partial f}{\partial x}=\,{3x^{2}+2xy,}\;\mbox{while}\;\frac{\partial f}% {\partial y}={x^{2}+2y.}