Basics of Partial Differentiation

In mathematics, sometimes the function depends on two or more than two variables. In this case, the derivative converts into the partial derivative since the function depends on several variables. Whereas, partial differential equation, is an equation containing one or more partial derivatives is called a partial differential equation. Partial derivatives are usually used in vector calculus and differential geometry. In this article students will learn the basics of partial differentiation.

Partial Derivative Rules

Just like ordinary derivatives, partial derivatives follows some rule like product rule, quotient rule, chain rule etc.

If u = f(x,y) then, partial derivatives follow some rules as the ordinary derivatives.

Product Rule:

If u = f(x,y).g(x,y), then

uxu_{x} = ux\frac{\partial u}{\partial x} = g(x,y)g\left ( x,y \right )fx\frac{\partial f}{\partial x}+f(x,y) + f\left ( x,y \right )gx\frac{\partial g}{\partial x}

And, uyu_{y} = uy\frac{\partial u}{\partial y} = g(x,y)g\left ( x,y \right )fy\frac{\partial f}{\partial y}+f(x,y) + f\left ( x,y \right )gy\frac{\partial g}{\partial y}

Quotient Rule:

If u = f(x,y)g(x,y)\frac{f(x,y)}{g(x,y)}, where g(x,y) \neq 0 then,

uxu_{x} = g(x,y)fxf(x,y)gx[g(x,y)]2\frac{g\left ( x,y \right )\frac{\partial f}{\partial x}-f\left ( x,y \right )\frac{\partial g}{\partial x}}{\left [ g\left ( x,y \right ) \right ]^{2}}

And, uyu_{y} = g(x,y)fyf(x,y)gy[g(x,y)]2\frac{g\left ( x,y \right )\frac{\partial f}{\partial y}-f\left ( x,y \right )\frac{\partial g}{\partial y}}{\left [ g\left ( x,y \right ) \right ]^{2}}

Power Rule:

If u = [f(x,y)]2 then, partial derivative of u with respect to x and y defined as

ux=n[f(x,y)]n1u_{x} = n\left [ f\left ( x,y \right ) \right ]^{n – 1} fx\frac{\partial f}{\partial x}

And, uy=n[f(x,y)]n1u_{y} = n\left [ f\left ( x,y \right ) \right ]^{n – 1} fy\frac{\partial f}{\partial y}

Chain Rule:

If u = f(x,y) is a function where, x = (s,t) and y = (s,t) then by the chain rule, we can find the partial derivatives us and ut as:

usu_{s} = ux.xs+uy.ys\frac{\partial u}{\partial x}.\frac{\partial x}{\partial s} + \frac{\partial u}{\partial y}.\frac{\partial y}{\partial s}

and utu_{t} = ux.xt+uy.yt\frac{\partial u}{\partial x}.\frac{\partial x}{\partial t} + \frac{\partial u}{\partial y}.\frac{\partial y}{\partial t}

Partial Derivative Examples

Given below are some of the examples on Partial Derivatives.

Question 1: Determine the partial derivative of a function fx and fy: if f(x, y) is given by f(x, y) = tan(xy) + sin x

Solution:

Given function is f(x, y) = tan(xy) + sin x

Derivative of a function with respect to x is given as follows:

fx = fx\frac{\partial f}{\partial x} = x\frac{\partial}{\partial x}[tan(xy)+sinx][\tan (xy) + \sin x]

= x\frac{\partial}{\partial x}[tan(xy)]+ [\tan(xy)] + x\frac{\partial}{\partial x} [sinx][\sin x]

= sec2 (xy). y + cos x

= y sec2(xy) + cos x

Now, Derivative of a function with respect to y. So, x is constant

fy = fy\frac{\partial f}{\partial y} = y\frac{\partial}{\partial y}[tan(xy)+sinx] [\tan (xy) + \sin x]

= y\frac{\partial}{\partial y}[tan(xy)]+ [\tan (xy)] + y\frac{\partial}{\partial y}[sinx][\sin x]

= sec2 (xy) x + 0

= x sec2 (xy)

Answer: fx = y sec2(xy) + cos x and fy = x sec2 (xy)

Question 2: If f(x,y) = 2x + 3y, where x = t and y = t2. Derivative f with respect to t.

Solution:

We know, dfdt=fxdxdt+fydydt\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}

Then, fx\frac{\partial f}{\partial x} = 2, fy\frac{\partial f}{\partial y} = 3, dxdt\frac{dx}{dt} = 1, dydt\frac{dy}{dt} = 2t

So, we get

dfdt\frac{df}{dt} = 2.1 + 3.(2t) = 2 + 3t

Question 3: If f=x2(yz)+y2(zx)+z2(xy)f = x^{2}(y – z) + y^{2}(z – x) + z^{2}(x – y), prove that fx\frac {\partial f} {\partial x} + fy\frac {\partial f} {\partial y} + fz\frac {\partial f} {\partial z}+0 + 0

Solution:

Given, f=x2(yz)+y2(zx)+z2(xy)f = x^{2} (y – z) + y^{2}(z – x) + z^{2}(x – y)

To find fx\frac {\partial f} {\partial x} ‘y and z’ are held constant and the resulting function of ‘x’ is differentiated with respect to ‘x’

fx\frac {\partial f} {\partial x} = 2x(yz)+0(01)+0(10)2x(y – z) + 0(0 – 1) + 0(1 – 0) fx\frac {\partial f} {\partial x} = 2xy2xz+0+02xy – 2xz + 0 + 0 fx\frac {\partial f} {\partial x} = 2xy2xz2xy – 2xz——–(i)

To find fy\frac {\partial f} {\partial y} ‘x and z’ is held constant and the resulting function of ‘y’ is differentiated with respect to ‘y’.

fy\frac {\partial f} {\partial y} = 0(1z)+2y(zx)+0(01)0(1 – z) + 2y(z – x) + 0(0 – 1) fy\frac {\partial f} {\partial y} = 0+2yz2yx+00 + 2yz – 2yx + 0 fy\frac {\partial f} {\partial y} = 2yz2yx2yz – 2yx ——-(ii)

To find fz\frac {\partial f} {\partial z} ‘x and y’ is held constant and the resulting function of ‘z’ is differentiated with respect to ‘z’.

fz\frac {\partial f} {\partial z} = 0(01)+0(11)+2z(xy)0(0 – 1) + 0(1 – 1) + 2z(x – y) fz\frac {\partial f} {\partial z} = 0+0+2zx2zy0 + 0 + 2zx – 2zy fz\frac {\partial f} {\partial z} = 2zx2zy2zx – 2zy ———(iii)

Adding (i) , (ii) and (iii) we get

fx\frac {\partial f} {\partial x} + fy\frac {\partial f} {\partial y} + fz\frac {\partial f} {\partial z} = 2xy2xz+2yz2yx2zx2zy2xy – 2xz + 2yz – 2yx – 2zx – 2zy fx\frac {\partial f} {\partial x} + fy\frac {\partial f} {\partial y} + fz\frac {\partial f} {\partial z} = 0

Hence proved.

Question 4: Given F = sin (xy). Show that ∂2F / (∂x ∂y) is equal to ∂2F / (∂y ∂x).

Solution:

sin(xy)

F / cos (xy)

F / cos (xy)

∂^2F / ∂ / [cos (xy)]

∂^2F / xsin (xycos (xy)

∂^2F / ∂ / [cos (xy)]

∂^2F / xsin (xycos (xy)

Hence,

2F / (∂x ∂y) =∂2F / (∂y ∂x)

Question 5: f (x, y) = x^2 + xy + y^2 , x = uv, y = u/v. To show that ufu + vfv = 2xfx and ufu − vfv = 2yfy

Solution: We need to find fu, fv, fx and fy.

fu = ∂f / ∂u = [∂f/ ∂x] [∂x / ∂u] + [∂f / ∂y] [∂y / ∂u];

fv = ∂f / ∂v = [∂f / ∂x] [∂x / ∂v] + [∂f / ∂y] [∂y / ∂v].

fu = (2x + y)(v) + (x + 2y)(1 / v) = 2uv^2 + 2u + 2u / v^2 .

fv = (2x + y)(u) + (x + 2y)(−u / v^2 ) = 2u^2 v − 2u^2 / v^3 .

fx = 2x + y = 2uv + u / v .

So, 2xfx = 2uvfx = 4u^2v^2 + 2u^2 .

fy = x + 2y = uv + 2u / v .

So, 2yfy = [2u / v] fx = 2u^2 + 4u^2 /  v^2 .

Now ufu + vfv = 2u^2 v^2 + 2u^2 + 2u^2 / v^2 + 2u^2 v^2 − 2u^2 / v^2

= 4u^2 v^2 + 2u^2 = 2xfx as required,

and ufu − vfv = 2u^2 v^2 + 2u^2 + 2u^2 / v^2 − 2u^2 v^2 + 2u^2 / v^2

= 2u^2 + 4u^2 / v^2 = 2yfy as required.

Question 6: Show that the largest triangle of the given perimeter is equilateral.

Solution: 

A = s (s − a) (s − b) (s − c)

Where

P = given

s = [1 / 2] P

a+b+c=P

c=P–a–b ← equation (1)

A=12P(12Pa)(12Pb)[12P(Pab)]A=12P(12Pa)(12Pb)(a+b12P)Aa=12P(12Pb)[(12Pa)(1)+(a+b12P)(1)]212P(12Pa)(12Pb)(a+b12P)=012Pa=a+b12PA = \sqrt{\frac{1}{2}P(\frac{1}{2}P – a)(\frac{1}{2}P – b)[\frac{1}{2}P – (P – a – b)]}\\ A = \sqrt{\frac{1}{2}P(\frac{1}{2}P – a)(\frac{1}{2}P – b)(a + b – \frac{1}{2}P)}\\ \frac{\partial A}{\partial a} = \frac{\frac{1}{2}P(\frac{1}{2}P – b)\left[ (\frac{1}{2}P – a)(1) + (a + b – \frac{1}{2}P)(-1) \right]}{2\sqrt{\frac{1}{2}P(\frac{1}{2}P – a)(\frac{1}{2}P – b)(a + b – \frac{1}{2}P)}} = 0\\\frac{1}{2}P – a = a + b – \frac{1}{2}P\\

b = P – 2a ←  equation (2)

Ab=12P(12Pa)[(12Pb)(1)+(a+b12P)(1)]212P(12Pa)(12Pb)(a+b12P)=012Pb=a+b12P\frac{\partial A}{\partial b} = \frac{\frac{1}{2}P(\frac{1}{2}P – a)\left[ (\frac{1}{2}P – b)(1) + (a + b – \frac{1}{2}P)(-1) \right]}{2\sqrt{\frac{1}{2}P(\frac{1}{2}P – a)(\frac{1}{2}P – b)(a + b – \frac{1}{2}P)}} = 0\\\frac{1}{2}P – b = a + b – \frac{1}{2}P\\

a = P – 2b

Substitute b = P – 2a of equation (2)

a=P2(P2a)a=P2P+4a3a=Pa=13Pa = P – 2(P – 2a)\\ a = P – 2P + 4a\\ 3a = P\\ a = \frac{1}{3}P\\

From equation (2)

b=P2(13P)b=13Pb = P – 2(\frac{1}{3}P)\\ b = \frac{1}{3}P\\

From equation (1)

c=P13P13Pc=13P Hence, a = b = c = P/3.c = P – \frac{1}{3}P – \frac{1}{3}P\\ c = \frac{1}{3}P\\ \text \ Hence, \ a \ = \ b \ = \ c \ = \ P/3.  Thus, the triangle is equilateral. The maximum area of the triangle is  :Amax=12P(12P13P)(12P13P)(12P13P)Amax=12P(16P)(16P)(16P)Amax=1432P4Amax=1123P2\text \ Thus, \ the \ triangle \ is \ equilateral.\\ \text \ The \ maximum \ area \ of \ the \ triangle \ is\ \ : A_{max} = \sqrt{\frac{1}{2}P(\frac{1}{2}P – \frac{1}{3}P)(\frac{1}{2}P – \frac{1}{3}P)(\frac{1}{2}P – \frac{1}{3}P)}\\ A_{max} = \sqrt{\frac{1}{2}P(\frac{1}{6}P)(\frac{1}{6}P)(\frac{1}{6}P)}\\ A_{max} = \sqrt{\frac{1}{432}P^4}\\ A_{max} = \frac{1}{12\sqrt{3}}P^2\\

Also read

Derivative examples

Differentiation of functions