The derivative of composite functions

جامعة بابل / كلية التربية للعلوم الصرفة / قسم الرياضيات 11 ( رياضيات ) 20
1
Derivatives
The derivative of the function y = f (x) is the function y? = f ?(x) Whose value at each x is define by
rule ( ) ( ) lim ( )
0
y f x
x
f x x
dx
y f x dy
x
= ? = ?
?
+ ?
= ? =
? ?
?? The Rules for Derivative
1
If y = b o
dx
? dy = where b is constant = 4 ? = 0
dx
y a dy
2
If y = xn ? = nxn?1
dx
dy
n any number = ?2 ? = ?2x?2?1 = ?2x?3
dx
y x dy
3 If = n ? = b.nxn?1
dx
y bx dy
3 2
1
3
1
3
3.
4
3
4. 4.1
x
x
dx
y = x ?dy = =
?
4 If y = u(x) +v(x)
dx
dv
dx
du
dx
? dy = + 2 2 8 5 4 4x 0 20x3
dx
y = x + ? x ? dy = + ?
5 If [ ] [ ]
dx
b n u x du
dx
y = b u(x) n ? dy = . ( ) n?1. = 3(2 2 ? + 4)7 ? = 3.7(2x2 ? x + 4)6.(4x ?1)
dx
y x x dy
6 If y = u(x).v(x)
dx
v x du
dx
u x dv
dx
? dy = ( ). + ( ). y = (x2 +1)(x ? 3)2
(x2 1)[2(x 3)] (x 3)2(2x)
dx
? dy = + ? + ?
7
if
( )
( )
v x
y = u x [ ( )]2
( ). ( ).
v x
dx
u x dv
dx
v x du
dx
dy ?
? = 2 2
2 2
(3 2 6)
( 1)
? +
+
=
x x
y x
جامعة بابل / كلية التربية للعلوم الصرفة / قسم الرياضيات 11 ( رياضيات ) 20
2
The derivative of composite functions ( Chain rule )
If y is differentiable function of (u ) and (u ) is differentiable function of ( x )
Then y is a differentiable function of ( x )
That is
du
y = f (u)? dy
dx
u = f (x)? du
dx
du
du
dy
dx
? dy = .
Find
dt
dy where y = x2 + x and x = 3t2 ? 2t +1
Solution:?
dt
dx
dx
dy
dt
dy = . ).(6 2)
2
? = (2 + 1 t ?
x
x
dt
dy
Substitute: x = 3t2 ? 2t + 6
](6 2)
2 3 2 6
[2(3 2 6) 1
2
2 ?
? +
= ? + + t
t t
t t
dx
dy
Find
dx
dy where x = 2t + 3 and y = t2 ?1
Solution:?
2
3
2
2 ?
= = t = t = x
dt
dx
dt
dy
dx
dy
جامعة بابل / كلية التربية للعلوم الصرفة / قسم الرياضيات 11 ( رياضيات ) 20
3
H.W :
1. Find
dt
dy by the chain rule expressing the results in terms of t :
, 2 5
2
y = x2 + x x = t ? ,
t
y = x + 2 , x = 2 , , 2 1
1
2
3
2
= + +
+
= x t t
x
y x
2. Find
dx
dz if z = w2 ? w?1 , w = 3x
3. Find
db
da if
b
a = 7r3 ? 2 , r =1? 1
4. Find
dt
dr if r (s 1)2 , s 10t 2 2t
1
= + = ?
5. Find
dt
dy if x = 3t +1 , y = t?3 + t