MA101.18 Partial Differentiation

Consider the curve in which the surface z=f(x,y) meets the plane y=c a constant.

In this plane z=f(x,c), and \displaystyle \frac{dz}{dx} would be a formula for the gradient of the tangent to the curve.

If we differentiate f(x,y) with respect to x, treating the y as if it was a constant (some say holding y constant), the the derivative obtained is called the partial derivative of f(x,y) with respect to x and we write \displaystyle \frac{\partial f}{\partial x} or f_x or D_x f. Similarily we have \displaystyle \frac{\partial f}{\partial y}.

With functions of more than two variables one differentiates with respect to one of the variables by holding all the other variables constant.

Examples

\displaystyle \frac{\partial}{\partial x} (x^2 y^2 + \tan{x}) = 2xy^2 + \sec{^2}{x}.

\displaystyle \frac{\partial}{\partial y}( x^2 y^2 + tan x) =2x^2 y

\displaystyle \frac{\partial}{\partial x} (2x+\sin{xy}) = 2 + y \cos{xy}

\displaystyle \frac{\partial}{\partial y} (2x+\sin{xy}) = x \cos{xy}

In the obvious way we can have higher order partial derivatives.

\displaystyle \begin{aligned} f(x,y) &= x^2 y + \sin{x} \\ \frac{\partial ^2 f}{\partial x^2} &= \frac{\partial}{\partial x}\bigg[\frac{\partial}{\partial x}(x^2y + \sin{x})\bigg] \\ &= \frac{\partial}{\partial x}\bigg[2xy+\cos{x}\bigg] \\ &= 2y - \sin{x}\end{aligned}

\displaystyle \begin{aligned} \frac{\partial ^2 f}{\partial y \partial x} &= f_{yx} \\ &= \frac{\partial}{\partial y}\bigg[\frac{\partial}{\partial x}(x^2y + \sin{x})\bigg] \\ &= 2x \end{aligned}

\displaystyle \begin{aligned} \frac{\partial ^2 f}{\partial x \partial y} &= f_{xy} \\ &= \frac{\partial}{\partial x}\bigg[\frac{\partial}{\partial y}(x^2y + \sin{x})\bigg] \\ &= 2x \end{aligned}

\displaystyle \begin{aligned} \frac{\partial ^2 f}{\partial y^2} &= 0 \end{aligned}

Similarly for V = \pi r^2 h we have:

\displaystyle \frac{\partial ^2 V}{\partial r^2} = V_{rr} = 2 \pi h

\displaystyle \frac{\partial ^2 V}{\partial r \partial h} = V_{rh} = 2 \pi r

\displaystyle \frac{\partial ^2 V}{\partial h \partial r} = V_{hr} = 2 \pi r

\displaystyle \frac{\partial ^2 V}{\partial h^2} = V_{hh} = 0

For commonly encountered functions f_{xy} we have that

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

From here on we may assume that all the mixed derivatives are equal.

Note, the normal rules (sum, product, quotient, function of a function) of differentiation apply to partial differentiation.

 

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