Partial Differentiation

The idea

For a function of several variables like $f(x,y)$ , a partial derivative with respect to one variable treats every other variable as a constant:

\dfrac{\partial f}{\partial x} = \lim_{h\to0}\dfrac{f(x+h,y)-f(x,y)}{h}

Worked example: first partial derivatives

For $f(x,y) = x^3y^2 + \sin(xy)$ :

\dfrac{\partial f}{\partial x} = 3x^2y^2 + y\cos(xy), \qquad \dfrac{\partial f}{\partial y} = 2x^3y + x\cos(xy)

The $y$ in $y\cos(xy)$ comes from the chain rule: the derivative of $xy$ with respect to $x$ is $y$ .

Worked example: mixed partials agree

Differentiating $\dfrac{\partial f}{\partial x}$ with respect to $y$ , and $\dfrac{\partial f}{\partial y}$ with respect to $x$ , both give:

\dfrac{\partial^2 f}{\partial y\,\partial x} = \dfrac{\partial^2 f}{\partial x\,\partial y} = 6x^2y + \cos(xy) - xy\sin(xy)

Worked example: gradient and directional derivative

For $g(x,y) = x^2y + y^3$ at $(1,2)$ : $g_x = 2xy = 4$ and $g_y = x^2 + 3y^2 = 13$ , so $\nabla g = (4, 13)$ . The rate of change in the direction of $(3,4)$ , using the unit vector $\left(\tfrac{3}{5}, \tfrac{4}{5}\right)$ , is:

D_{\hat{u}}g = 4\left(\tfrac{3}{5}\right) + 13\left(\tfrac{4}{5}\right) = \dfrac{64}{5} = 12.8

The largest possible rate of change is $|\nabla g| = \sqrt{16 + 169} = \sqrt{185} \approx 13.6$ , in the direction of $\nabla g$ itself.

Worked example: a minimum of two variables

Find the extreme value of $h(x,y) = x^2 + y^2 - 2x - 4y$ .

Solution: $h_x = 2x - 2 = 0$ and $h_y = 2y - 4 = 0$ give the critical point $(1, 2)$ . With $r = h_{xx} = 2$ , $s = h_{xy} = 0$ and $t = h_{yy} = 2$ , $rt - s^2 = 4 > 0$ and $r > 0$ , so it's a minimum: $h(1,2) = 1 + 4 - 2 - 8 = -5$ .

Common mistakes

Treating the other variable as a function. In $\partial/\partial x$ , $y$ is a constant: $\partial(x^2y)/\partial x = 2xy$ .
Forgetting the chain-rule factor in terms like $\sin(xy)$ .
Using a non-unit direction vector. Divide by its length before taking the directional derivative.
Calling every critical point an extremum. If $rt - s^2 < 0$ it's a saddle point; if it's 0 the test is inconclusive.

Where this comes up in exams

Partial derivatives and maxima and minima are listed under Calculus in the GATE 2027 EC, EE and IN syllabi, and the EE syllabus also lists directional derivatives.

Checked against GATE 2027 EC syllabus, GATE 2027 EE syllabus and GATE 2027 IN syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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