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Mathematics

Mathematics 0110A/B

Chris Brandl

Fall

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MATH 127 Fall 2012
Assignment 6 Solutions
1. Compute the indicated derivatives for the following functions.
2
6x ▯ x 0
(a) f(x) = 2 p , f (x)
x + x
▯ ▯
▯ 2 2 p 2▯ 1
6 + x2)(x + x) ▯ (6x ▯ x 2x + 2 x
f (x) = p
(x + x)2
▯ p ▯ ▯ p ▯
6x + 2 + 6 x + 2 ▯ 12x ▯ 4 + 3 x ▯ 1
x2 x2
= 2 p 2
(x + x)
▯6x + 3 px + 3 + 6
x2
= 2 p 2
(x + x)
2
x 2 00
(b) f(x) = ▯ 2, f (x)
2 x
0 4
f (x) = x + 3
x
00 12
f (x) = 1 ▯ 4
x
x +1 2 0
(c) f(x) = e ln(x + 1), f (x)
▯ ▯
f (x) = (2x)ex +1 ln(x + 1) + ex +1 2x
x + 1
cos(▯ x)
(d) f(x) = , f (x)
sin(x)
p p
▯ p sin(▯ x)(sin(x)) ▯ cos(▯ x)cos(x)
f (x) = 2 x
(sin(x))2
2. Find f (x)
▯ 3
(a) f(x) = x arctan(sin(x ))
2 3
f (x) = ▯x ▯▯1arctan(sin(x )) + x▯ 3x cos(x )
1 + (sin(x ))
1 ▯ ▯
1 ln(x)
(b) f(x) =
x
We will use logarithmic di▯erentiation to ▯nd the indicated derivative. We ▯rst
take the logarithm of both sides of the equation and simplify.
▯ ▯
1 ln(x)
y =
x
▯ ▯ ln(x)
1
ln(y) = ln x
▯ 1
= ln(x)ln
x
= ▯(ln(x))2
Di▯erentiating both sides we get
y0 2ln(x)
= ▯
y x
0 2ln(x)
y = ▯ y
x ▯ ▯
2ln(x) 1 ln(x)
= ▯
x x
Thus, we have
▯ ▯ ln(x)
0 2ln(x) 1
f (x) = ▯ x x
x
(c) f(x) = (cos(x))
We will compute the derivative of (cos(x)) using logarithmic di▯erentiation.
y = (cos(x))
x
ln(y) = ln((cos(x)) )
= xln(cos(x))
▯ ▯
y0 sin(x)
= ln(cos(x)) + x ▯
y cos(x)
0
y = (ln(cos(x)) ▯ xtan(x))y
= (ln(cos(x)) ▯ xtan(x))(cos(x))x
2 Thus, we have
0 x
f (x) = (ln(cos(x)) ▯ xtan(x))(cos(x))
p
arcsin(x) cos(x) + 2
(d) f(x) = x 3 , ▯nd the derivative without using the product or
e (x + x)
quotient rules.
p
arcsin(x) cos(x) + 2
We will use logarithmic di▯erentiation on y = e (x + x) . We ▯rst
take the logarithm of both sides and simplify.
p !
arcsin(x) cos(x) + 2
lny = ln 2
e (x + x)
▯ ▯ ▯ ▯
p x 3
= ln arcsin(x) cos(x) + 2 ▯ ln e (x + x)
▯p ▯ ▯ ▯ ▯ ▯
= ln(arcsin(x)) + ln cos(x) + 2 ▯ ln e x2 ▯ ln x + x
▯ ▯
= ln(arcsin(x)) +ln(cos(x) + 2)▯ x ▯ ln x + x
2
0 ▯ ▯ 2
y = 1 p 1 ▯ sin(x) ▯ 2x ▯ 3x + 1
y arcsin(x) 1 ▯ x2 2(cos(x) + 2) x + x
▯ ▯ ▯ 2 ▯
0 1 1 sin(x) 3x + 1
y = p 2 ▯ ▯ 2x ▯ 3 y
arcsin(x) 1 ▯ x 2(cos(x) + 2) x + x !
▯ ▯ ▯ 2 ▯ p
= 1 p 1 ▯ sin(x) ▯ 2x ▯ 3x + 1 arcsin(x) cos(x) + 2
arcsin(x) 1 ▯ x2 2(cos(x) + 2) x + x e (x + x)
3. For each of the following equations, ▯nd the indicated derivative.
4y
(a) 4x + y ▯ = 10x , ▯nd y .
x + 3
We perform implicit di▯erentiation
0
2 0 4y (x + 3) ▯ 4y

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