1. Prove the Derivative Product Rule? d dv du + v dx %3D u- dr (uv) dx

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
KB/S
Solution:
Proof
[u(x + h) + v(x + h)] – [u(x) + v(x)]
dr [u(x) + v(x)]
= lim
h→0
h
и(х + h) — и(х)
= lim
v(x + h) – v(x)
h
h
u(x + h) – u(x)
lim
v(x + h) – v(x)
+ lim
du
dx
dv
%3D
h→0
h
h0
h
dx
Differentiation Rules
Home Work :
1. Prove the Derivative Product Rule?
d
dv
du
dx
+ v
(uv) = u
dx
dx
2. Prove the Derivative Quotient Rule?
du
dx
dv
d
dx
и
dx
Differentiation Rules
Example 7: Differentiate the following functions of x.
(a) x³
1
(d)
„2+7
(b) x/3
(c) xV?
(e) x~4/3
(f)
(g) 10
Solution:
(а)
dx
:(x³) = 3x³=1 = 3x²
(b) 4 (12#) = e/-1 =
d
dr (xV2) = V2rV2-1
(c)
%3D
d
(d)
dx
d
4
(x¬4) = -4x¬4-1
-4x-5
%3D
=
= -
dx
(e) (-/3) = -4/3)–1 = --73
m (VFF) = ) = (1 + )*/-1 = 2 + m)VF
(B)을 (10)3 0
dx
d
(f)
dx
d
yl+(#/2)-1 =
dx
d
dx
+
Transcribed Image Text:KB/S Solution: Proof [u(x + h) + v(x + h)] – [u(x) + v(x)] dr [u(x) + v(x)] = lim h→0 h и(х + h) — и(х) = lim v(x + h) – v(x) h h u(x + h) – u(x) lim v(x + h) – v(x) + lim du dx dv %3D h→0 h h0 h dx Differentiation Rules Home Work : 1. Prove the Derivative Product Rule? d dv du dx + v (uv) = u dx dx 2. Prove the Derivative Quotient Rule? du dx dv d dx и dx Differentiation Rules Example 7: Differentiate the following functions of x. (a) x³ 1 (d) „2+7 (b) x/3 (c) xV? (e) x~4/3 (f) (g) 10 Solution: (а) dx :(x³) = 3x³=1 = 3x² (b) 4 (12#) = e/-1 = d dr (xV2) = V2rV2-1 (c) %3D d (d) dx d 4 (x¬4) = -4x¬4-1 -4x-5 %3D = = - dx (e) (-/3) = -4/3)–1 = --73 m (VFF) = ) = (1 + )*/-1 = 2 + m)VF (B)을 (10)3 0 dx d (f) dx d yl+(#/2)-1 = dx d dx +
KB/S
Solution:
Proof
[u(x + h) + v(x + h)] – [u(x) + v(x)]
dr [u(x) + v(x)]
= lim
h→0
h
и(х + h) — и(х)
= lim
v(x + h) – v(x)
h
h
u(x + h) – u(x)
lim
v(x + h) – v(x)
+ lim
du
dx
dv
%3D
h→0
h
h0
h
dx
Differentiation Rules
Home Work :
1. Prove the Derivative Product Rule?
d
dv
du
dx
+ v
(uv) = u
dx
dx
2. Prove the Derivative Quotient Rule?
du
dx
dv
d
dx
и
dx
Differentiation Rules
Example 7: Differentiate the following functions of x.
(a) x³
1
(d)
„2+7
(b) x/3
(c) xV?
(e) x~4/3
(f)
(g) 10
Solution:
(а)
dx
:(x³) = 3x³=1 = 3x²
(b) 4 (12#) = e/-1 =
d
dr (xV2) = V2rV2-1
(c)
%3D
d
(d)
dx
d
4
(x¬4) = -4x¬4-1
-4x-5
%3D
=
= -
dx
(e) (-/3) = -4/3)–1 = --73
m (VFF) = ) = (1 + )*/-1 = 2 + m)VF
(B)을 (10)3 0
dx
d
(f)
dx
d
yl+(#/2)-1 =
dx
d
dx
+
Transcribed Image Text:KB/S Solution: Proof [u(x + h) + v(x + h)] – [u(x) + v(x)] dr [u(x) + v(x)] = lim h→0 h и(х + h) — и(х) = lim v(x + h) – v(x) h h u(x + h) – u(x) lim v(x + h) – v(x) + lim du dx dv %3D h→0 h h0 h dx Differentiation Rules Home Work : 1. Prove the Derivative Product Rule? d dv du dx + v (uv) = u dx dx 2. Prove the Derivative Quotient Rule? du dx dv d dx и dx Differentiation Rules Example 7: Differentiate the following functions of x. (a) x³ 1 (d) „2+7 (b) x/3 (c) xV? (e) x~4/3 (f) (g) 10 Solution: (а) dx :(x³) = 3x³=1 = 3x² (b) 4 (12#) = e/-1 = d dr (xV2) = V2rV2-1 (c) %3D d (d) dx d 4 (x¬4) = -4x¬4-1 -4x-5 %3D = = - dx (e) (-/3) = -4/3)–1 = --73 m (VFF) = ) = (1 + )*/-1 = 2 + m)VF (B)을 (10)3 0 dx d (f) dx d yl+(#/2)-1 = dx d dx +
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