Partial derivatives

Partial derivatives & multiple integrals

Japanese school year: University year 1

What you learn

You will learn how to compute partial derivatives of multivariable functions by differentiating with respect to one variable while treating others as constants. This method is fundamental in solving multivariable optimization problems and analyzing physical fields. Mastery of single-variable differentiation rules is an essential prerequisite.

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Key points

This defines the partial derivative of $f(x, y)$ with respect to $x$ . It measures the rate of change when varying only $x$ while holding $y$ constant.

\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}

This equation states that the order of partial differentiation does not matter: differentiating by $y$ then $x$ equals differentiating by $x$ then $y$ (valid for smooth functions).

\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}

This chain rule finds the total rate of change $dz/dt$ when both $x$ and $y$ depend on $t$ . It sums the rate transmitted through $x$ and the rate transmitted through $y$ .

\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}

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Practice 1Coming soon