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Partial Derivatives

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الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 1
أستاذ المادة فؤاد حمزة عبد الشريفي       07/03/2017 15:01:07
Partial Derivatives
For a function of two independent variables, f(x,y), the partial derivative of f
with respect to x can be found by applying all the usual rules of differentiation. The only exception is that, whenever and wherever the second variable y appears, it is treated as a constant in every respect. The partial derivative of f with respect to y
can similarly be found by treating x as a constant whenever it appears.
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.
Notations of partial derivatives:
Partial derivative of f with respect to x ?f/?x f_x
Partial derivative of f with respect to y ?f/?y f_y
Example 1: Find f_x and f_y for f(x,y)=x^2 y^4
f_x=2xy^4 and f_y=4x^2 y^3
Example 2: Find f_x and f_y for f(x,y)=x^3+y^2
f_x=3x^2 and f_y=2y
Example 3: Find w_x and w_y for w(x,y)=x^2 cos?(xy)
w_x=x^2 (-sin?(xy) )×y+2x cos?(xy)=-x^2 y sin?(xy)+2x cos?(xy)
w_y=x^2 (-sin?(xy) )×x=-x^3 sin?(xy)
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)

Higher Order Partial Derivatives
Let z=f(x,y) be a function of x and y.
The second partial derivative of f with respect to x (?^2 f)/(?x^2 ) or (?^2 z)/(?x^2 ) f_xx or z_xx
The second partial derivative of f with respect to y (?^2 f)/(?y^2 ) or (?^2 z)/(?y^2 ) f_yy or z_yy
The second partial derivative of f with respect to y and then with respect to x. (?^2 f)/?x?y or (?^2 z)/?x?y f_xy or z_xy
The second partial derivative of f with respect to x and then with respect to y. (?^2 f)/?y?x or (?^2 z)/?y?x f_yx or z_yx

f_xy and f_yx are called mixed partial derivatives and f_xy=f_yx
Example 5: 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 6: 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??
f_?=r^2 sin?2?
f_??=2r^2 cos?2?





Laplace s Equation
We say that the function z=f(x,y) satisfies Laplace s equation if
(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )=0
Example 7: 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 ? (?^2 f)/(?x^2 )=-4e^(-2y) cos?2x
?f/?y=-2e^(-2y) cos?2x ? (?^2 f)/(?y^2 )=4e^(-2y) cos?2x

(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )=-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




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
Example8: If z=x^2+y^2 and x=2u +?v , y=2v -?u find ?z/?u and ?z/?v
?z/?x=2x=4u+2v , ?z/?y=2y=4v-2u , ?x/?u=2 and ?y/?u=-1
?z/?u=?z/?x×?x/?u+?z/?y×?y/?u=(4u+2v )×2+(4v-2u )×(-1)
=8u+4v-4v+2u=10u
?x/?v=1 and ?y/?v=2
?z/?v=?z/?x×?x/?v+?z/?y×?y/?v=(4u+2v )×1+(4v-2u )×2
=4u+2v+8v-4u=10v


1.If w=cos?(x+y)+sin?(x-y)then show that (?^2 w)/(?x^2 )=(?^2 w)/(?y^2 ).
2.Find (?^2 z)/?x?y for z=x^2 tan^(-1)??( y )/( x )?.
3.Show that f(x,y)=e^3x sin??3x ? satisfy Laplace^ s equation.

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


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