Handout - Derivative - Chain Rule & Sin x Cos x ex ln(x Power-Chain ...
Handout - Derivative - Chain Rule & Sin(x), Cos(x), ex, ln(x)
Power-Chain Rule a,b are constants.
Function
Derivative
y = a ? xn
dy = a ? n ? xn-1 dx
y = a ? un
dy = a ? n ? un-1 ? du
dx
dx
Power Rule Power-Chain Rule
1
Exercises Find the derivatives of the expressions
a) 6(9x + 26)9
b) 6(8x + 29)8
c) 5 x2 + 3x + 23 9
d)5 7x2 + 2x + 22 6
e) 7 2x2 + 5x + 28 25/3
g) 4 4x2 + 3x + 26
f)8 5x2 + 7x + 23 16/3
h)6 2x2 + 5x + 27
i)
9 3x2+6x+29
j)
9 5x2+6x+25
Answers a) 486(9x + 26)8; b) 384(8x + 29)7; c) 45(2x + 3) x2 + 3x + 23 8; d) 30(14x + 2) 7x2 + 2x + 22 5;
e)
175 3
(4x
+
5)
2x2 + 5x + 28
22/3;
f)
128 3
(10x
+ 7)
5x2 + 7x + 23
13/3;
g)
2(8x+3) 4x2+3x+26
;
h)
3(4x+5) 2x2+5x+27
;
i)
-
9(6x+6) 2(3x2+6x+29)3/2
;
j)
-
9(10x+6) 2(5x2+6x+25)3/2
;
2
Sine and Cosine - Chain Rules a,b are constants.
Function y = sin(x) y = cos(x) y = a ? sin(u) y = a ? cos(u)
Derivative dy
= cos(x) dx
dy = - sin(x)
dx
dy
du
= a ? cos(u) ?
dx
dx
dy
du
= -a ? sin(u) ?
dx
dx
Sine Rule Cosine Rule Chain-Sine Rule Chain-Cosine Rule
3
Exercises Find the derivatives of the expressions
a) 7 cos 2x2 + 5x + 9
b) 8 cos 5x3 + 5x2 + 2
c) 5 cos x3 + 3x2 + 1
d) 7 cos 7x2 + 1
e) 6 sin 4x4 + 4 cos(3x) 3
f) 7 sin 9x2 + 5 cos 6x5 4
g) 8 sin 6x4 + cos 8x4 3
h) 4 sin 8x4 + cos 8x2 4
5
i) 7 sin(4x) + 6 cos 6x4 2
1
j) 7 sin(5x) + 9 cos 4x3 2
Answers a) -7(4x + 5) sin 2x2 + 5x + 9 ; b) -8 15x2 + 10x sin 5x3 + 5x2 + 2 ;
c) -5 3x2 + 6x sin x3 + 3x2 + 1 ; d) -98x sin 7x2 + 1 ;
e) 3 ? 96x3 cos 4x4 - 12 sin(3x) ? 6 sin 4x4 + 4 cos(3x) 2;
f) 4 ? 126x cos 9x2 - 150x4 sin 6x5 ? 7 sin 9x2 + 5 cos 6x5 3;
g) 3 ? 192x3 cos 6x4 - 32x3 sin 8x4 ? 8 sin 6x4 + cos 8x4 2;
h) 4 ? 128x3 cos 8x4 - 16x sin 8x2 ? 4 sin 8x4 + cos 8x2 3;
i)
5 2
?
28 cos(4x) - 144x3 sin
6x4
j)
1 2
?
35 cos(5x) - 108x2 sin
4x3
3
? 7 sin(4x) + 6 cos 6x4 2 ;
? 7 sin(5x) + 9 cos 4x3
-
1 2
;
4
Exponent and Logarithmic - Chain Rules a,b are constants.
Function y = ex y = ln(x) y = a ? eu y = a ? ln(u)
Derivative dy = ex dx
dy 1 =
dx x
dy = a ? eu ? du
dx
dx
dy a du =?
dx u dx
Exponential Function Rule Logarithmic Function Rule Chain-Exponent Rule Chain-Log Rule
5
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