Partial Derivatives

Partial Derivatives
Let f(x,y) be a function of two variables, then f_x is defined as the derivative
of the function f(x,y), where y is considered a constant. It is called partial derivative of f with respect to x. The partial derivative with respect to y is defined
similarly.
For a function of more than two independent variables, the same method applies. Its partial derivative with respect to x, can be obtained by differentiating it with respect to x, using all the usual rules of differentiation. Except that all the other independent variables, whenever and wherever they occur in the expression of f, are
treated as constants.
f_x ? ?f/?x means partial derivative of f with respect to x
f_y ? ?f/?y means partial derivative of f with respect to y
Example 1: Find f_x and f_y for 1. f(x,y)=x^2 y^4 2.f(x,y)=x^3+y^2
1. f_x=2xy^4 and f_y=4x^2 y^3
2. f_x=3x^2 and f_y=2y
Example 2: Find ?w/?x and ?w/?y for w(x,y)=x^2 cos?(y)
?w/?x=2x cos?(y) , ?w/?y=-x^2 sin?(y)
Example 3: Show that f(x,t)=e^(-(x+t)^2 ) satisfy the advection equation f_x=f_t.
f_x=-2(x+t) e^(-(x+t)^2 ) and f_t=-2(x+t) e^(-(x+t)^2 )



Example 4: Find h_s and h_t for h(s,t)=t ln?(4s^2+1)+t^2 tan^(-1)?(2s)
h_s=8st/(4s^2+1)+(2t^2)/(1+4s^2 )=(8st+2t^2)/(4s^2+1)
h_t=ln?(4s^2+1)+2t tan^(-1)?(2s)
Example 5: If f(x,y)=(x-y)/(x+y) ,then show that x ?f/?x +y ?f/?y=0
?f/?x=((x+y)-(x-y))/(x+y)^2 =2y/(x+y)^2
?f/?y=((x+y)-(x-y)×-1)/(x+y)^2 =(-2x)/(x+y)^2
x ?f/?x +y ?f/?y=2xy/(x+y)^2 +(-2xy)/(x+y)^2 =0
Example 6: If f(x,y,z)=x sin?(yz)+xe^yz ,then show that
1. f(x,y,z)=xf_x 2. yf_y=zf_z
1. f_x=sin?(yz)+e^yz
xf_x=x sin?(yz)+xe^yz=f(x,y,z)
2. f_y=xz cos?(yz)+xze^yz ? yf_y=xyz cos?(yz)+xyze^yz
f_z=xy cos?(yz)+xye^yz ? zf_z=xyz cos?(yz)+xyze^yz
So yf_y=zf_z






Second order partial derivatives
Let f(x,y) be a function of x and y, then the second partial derivative of f with respect to x is f_xx , the second partial derivative of f with respect to y is f_yy , the second partial derivative of f with respect to y and then with respect to x is f_xy and the second partial derivative of f with respect to x and then with respect to y is f_yx.
( f_xx?(?^2 f)/(?x^2 ) ,f_yy?(?^2 f)/(?y^2 ) ,f_xy?(?^2 f)/?x?y and f_yx?(?^2 f)/?y?x )
f_xy and f_yx are called mixed partial derivatives and f_xy=f_yx.
Example 7: Find f_xx ,f_xy ,f_yx and f_yy for f(x,y)=x^2+3xy-y^4
f_x=2x+3y ? f_xx=2 and f_yx=3
f_y=3x-4y^3 ? f_yy=-12y^2 and f_xy=3
Example 8: Find f_rr and f_?? for f(r,?)=r^2 sin^2??
f_r=2r sin^2?? ? f_rr=2 sin^2??
f_?=2r^2 sin?? cos?? but (2 sin?? cos??=sin?2? )
f_?=r^2 sin?2? ? f_??=2r^2 cos?2?
Example 9: Find all the first and second order partial derivatives of
f(x,y,z)=3x^2-2xy^2+4x^2 z+z^3 y+5
f_x=6x-2y^2+8xz ? f_xx=6+8z ,f_yx=-4y and f_zx=8x
f_y=-4xy+z^3 ? f_yy=-4x , f_xy=-4y and f_zy=2z^2
f_z=4x^2+2z^2 y ? f_zz=4zy , f_xz=8x and f_yz=2z^2





Laplace s Equation: We say that the function f(x,y) satisfies Laplace s equation if
(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )=0 (f_xx+f_yy=0)
Example 10: Show that the functions satisfy Laplace s equation
1.f(x,y)=e^(-2y) cos?2x 2.w(s,t)=ln?(t^2+s^2 )
1. f_x=-2e^(-2y) sin?2x ? f_xx=-4e^(-2y) cos?2x
f_y=-2e^(-2y) cos?2x ? f_yy=4e^(-2y) cos?2x

f_xx+f_yy=-4e^(-2y) cos?2x+4e^(-2y) cos?2x=0
2. ?w/?s=2s/(t^2+s^2 ) ? (?^2 w)/(?s^2 )=(2(t^2+s^2 )-2s×2s)/(t^2+s^2 )^2
(?^2 w)/(?s^2 )=(2t^2+2s^2-4s^2)/(t^2+s^2 )^2 =(2t^2-2s^2)/(t^2+s^2 )^2
?w/?t=2t/(t^2+s^2 ) ? (?^2 w)/(?t^2 )=(2(t^2+s^2 )-2t×2t)/(t^2+s^2 )^2
(?^2 w)/(?t^2 )=(2t^2+2s^2-4t^2)/(t^2+s^2 )^2 =(2s^2-2t^2)/(t^2+s^2 )^2

(?^2 w)/(?s^2 )+(?^2 w)/(?t^2 )=(2t^2-2s^2)/(t^2+s^2 )^2 +(2s^2-2t^2)/(t^2+s^2 )^2 =0
The wave equation u_tt-c^2 u_xx=0
Example 11: Show that u(x,t)=sin?(x+2t)-sin?(x-2t) satisfy the wave equation u_tt-4u_xx=0.
u_t=2 cos?(x+2t)+2 cos?(x-2t) ? u_tt=-4 sin?(x+2t)+4 sin?(x-2t)
u_x=cos?(x+2t)-cos?(x-2t) ? u_xx=-sin?(x+2t)+sin?(x-2t)
u_tt-4u_xx=-4 sin?(x+2t)+4 sin?(x-2t)-4[-sin?(x+2t)+sin?(x-2t) ]=0

Chain rule for partial derivatives
If z=f(x,y) and x=x(u,v) ,y=y(u,v) then z is a function of u and v and
?z/?u=?z/?x×?x/?u+?z/?y×?y/?u
?z/?v=?z/?x×?x/?v+?z/?y×?y/?v
Example 12: If z=x^2+y^2 , x=2u +?v , y=2v -?u.
Find ?z/?u and ?z/?v as a functions of u and v
?z/?x=2x=4u+2v and ?z/?y=2y=4v-2u
?x/?u=2 and ?y/?u=-1
?x/?v=1 and ?y/?v=2
?z/?u=?z/?x×?x/?u+?z/?y×?y/?u
=(4u+2v )×2+(4v-2u )×(-1)=8u+4v-4v+2u=10u
?z/?v=?z/?x×?x/?v+?z/?y×?y/?v
=(4u+2v )×1+(4v-2u )×2=4u+2v+8v-4u=10v


1.Show that w(x,t)=cos?(x+3t)+sin?(x-3t) satisfy the wave equation u_tt-9u_xx=0.
2.Find z_xy if z(x,y)=x^2 sin?(2x-3y).
3.Show that f(x,y)=x^3 -??3xy^2 ? satisfy Laplace^ s equation.

4.If z=ln?(xy) and x=r sin?? , y=r cos?? then show that ?z/??=2 cot?2? .