MATH 136
Chain Rule
Evaluate f ′(x) :
1. f (x) =
cos(4 x)
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cos(4 x)
1
−1 / 2
f ′(x) = ( cos(4 x) )
×
× (−sin(4 x)) × 4
2
cos(4 x)
−2 cos(4 x) tan(4 x)
=
= −2 cos(4 x) tan(4 x)
cos(4 x)
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€
2. f (x) = sec 3 (3x 2 )
(
)
f ′(x) = 3 sec(3x 2 )
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2
× sec(3x 2 )tan(3x 2 ) × 6x
= 18x sec 3 (3x 2 ) tan(3x 2 )
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3. f (x) = ln( sin x
)
f ′(x) =
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sin x
1
×
× cos x = cot x
sin x
sin x
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2
4. f (x) = e−4 tan(x )
2
f ′(x) = e−4 tan(x ) × −4 sec 2 (x 2 ) × 2x
(
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2
= −8x sec 2 (x 2 )e−4 tan(x )
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)