Unit VI- Differentiation
A. Partial Derivatives
B. Mean Value Theorems
C. Successive Differentiation
D. Approximation and Errors
E. Maxima and Minima
A. Partial Derivatives
Let z=f(x,y) be a function in two variables x and y.
∂z
∂x
∂z
∂y
= derivative of z wrt x keeping y constant.
= derivative of z wrt y keeping x constant.
Remarks:
1. Higher order partial derivatives:
1.
∂2 z
∂
∂x
∂x ∂x
∂z
= ( )
2
2.
∂2 z
∂
∂z
∂2 z
∂y
∂y ∂y
∂xy
=
2
( ) 3.
∂z
∂2 z
∂x ∂y
∂yx
∂
= ( ) 4.
=
∂
∂z
( )
∂y ∂x
2. Euler’s Theorem
Let z be a homogeneous function of two variables x and y of degree n then
I. x
∂z
+y
∂x
∂z
∂y
=nz
II. x2
∂2 z
∂x2
+2xy
∂2 z
∂xy
+y2
∂2 z
∂y2
=n(n-1)z
Exercise 6.1
Q.1 Find
∂z
∂x
and
∂z
∂y
of the following:
a. z= x3+4xy2+y3
e. z= x2y3+x/y
b. z= xy+yx
c. z= xsiny +ysinx d. z= log(x2+y2)
f. z= exycos(xy) g. z= tan-1(y/x)
h. z= 10(3x+4y)
Q.2 Find higher order partial derivatives of the following:
a. z= x5+4xy+y5 b. z=cos(xy) c. z= √x + y d. z= tan-1(xy)
Q.3 Verify Euler’s Theorem for the following:
a. z= x4+xy3+y4 b. z=3x2-5xy+4y2 c. z=√xy
Q.4 If z= xy+yx then verify that
∂2 z
∂2 z
=
.
∂x ∂y ∂y ∂x
∂2 z
∂2 z
Q.5 If z=tan-1(y/x), prove that ∂x2 +∂y2 =0.
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B. Mean Value Theorems:
1. Rolle’s Theorem
If f(x) is continuous on [a,b], differentiable on (a,b) and f(a)=f(b) then there exists
c∈(a,b) such that f ’(c)=0.
2. Lagrange’s Mean value theorem
If f(x) is continuous on [a,b] and differentiable on (a,b) then there exists c∈(a,b) such
that f ‘(c)=(f(b)-f(a))(b-a).
3. Cauchy Mean Value theorem
If f(x) and g(x) are continuous on [a,b] and differentiable on (a,b) then there exist
c∈(a,b) such that
f(b)−f(a)
g(b)−g(a)
=
f′ (c)
g′ (c)
Exercise 6.2
Q.1 Verify Rolle’s theorem for the following:
(a) f(x)=x2-3x+2 for [1,2]
(b) f(x)=(x-3)(x-7) for [3,7]
(c) f(x)=e-xsinx for [0,π]
(d) f(x)=x2-5x+7 on [0,5]
Q.2 Verify Lagrange’s theorem for the following:
(a) f(x)= (x-1)(x-2)(x-4) on [0,4]
(b) f(x)=2x2-7x+10 on [2,5]
(c) f(x)= logx on [1,e]
(d) f(x)= lx2+mx+n on [a,b]
Q.3 Verify CMVT.
(a) f(x)=3x+2, g(x)=x2+1 on [1,4]
(b) f(x)=sinx, g(x)=cosx on [0,π/2]
(c) f(x)=ex, g(x)=e-x on [0,1]
(d) f(x)=√x , g(x)=
1
√x
on [4,9]
C. Successive Differentiation
Let y=f(x).
n
d y
yn= nth order derivative of y ie f(x)= dxn = f(n)(x).
2
Sr. N.
y=f(x)
yn
1.
c=constant
0
2.
(ax+b)m
npm an(ax+b)m-n if n≤m
otherwise zero
(−1)n n! an
(ax + b)n+1
aneax
3.
1
ax + b
4.
eax
5.
sin(ax+b)
ansin(ax+b+ 2 )
6.
cos(ax+b)
ancos(ax+b+ )
7.
log(ax+b)
nπ
nπ
2
(−1)n−1 (n − 1)! an
(ax + b)n
Exercise 6.3
Find nth order derivative of the following:
1. y=
x
(x−1)(x−3)
2
2. y=(x−1)(x−2)(x−3)
3. y=
4. y=
4
x2 +1
x2
x+1
5. y=sin3x
6. y=cos2x
7. y= cos2x.cos5x
8. y= sinx.sin2x.sin3x
9. y=sin3x.sin5x
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D. Approximations and Errors
Let z=f(x,y) be a differentiable function.
Let δx, δy, δz be amount of error (or increment) in x, y, z respectively.
Then,
δz=
∂z
∂x
δx+
∂z
∂y
δy
Percentage error: If δx is an error in x then
δx
x
×100 is known as percentage error in x.
Exercise 6.4
Q.1 If f(x,y)=x2+y2+xy, find f(2.01,2.001) approximately.
Q.2 If f(x,y)=exy, compute f(1.1,2.01) approximately, given e2=7.389.
Q.3 Find percentage error in area of an ellipse of 1% error is made in measuring major
and minor axes of the ellipse.
l
Q.4 The period of simple pendulum is T=2π√g. Find the percentage error in T due to
possible errors of 1% in l and 2.5% in g.
Q.5 Find the possible percentage error in computing the parallel resistance R from two
resistances R1 and R2 if in both the resistances error of 2% is made.
Q.6 The sides of a triangle can be measured as 12cm and 15cm and included angle is
60o. If the possible errors in the sides are 1% and in the angle is 2% then find the
percentage error in determining the area.
Q.7 Find the approximate value of [(0.98)2+(2.01)2].
E. Maxima/Minima
Steps to find maxima/minima of f(x,y)
∂f
∂f
1. Solve ∂x =0 and ∂y =0 simultaneously.
Let one of the solution be (a,b)
∂2 f
∂2 z
∂2 z
2. Let ∂x2 =r , ∂y2 =s, ∂x ∂y =t at (a,b).
3. (a,b) gives maxima if rs-t2>0 and r<0.
4. (a,b) gives minima if rs-t2>0 and r>0.
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Exercise 6.5
Q.1 Find the extreme values of f(x,y)=x2y2(1-x-y).
Q.2 Find the extreme values of xy(1-x-y).
Q.3 Discuss the maxima and minima of x3+y3-9xy.
Q.4 Divide 640 into three parts such that the sum of their products taken at a time is
maximum.
Q.5 Find maxima and minima of x3+3xy2 – 15x2-15y2+72x.
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