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(Solved) Section 2. #18, #74 In problems 12-22, compute the derivative. sin(2 x) cos(5 x +1) d 18). dx In Problems 72-83, use the differentiation patterns 1 1...


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Section 2.4: #18, #74
In problems 12–22, compute the derivative.
sin(2 x) ∙ cos(5 x +1)
d
18).
¿
dx In Problems 72–83, use the differentiation patterns
1
1
1
D ( artan ( x ) )=
, D ( arcsin ( x ) )=
∧D ( ln ( x ) )=
2
2
x
1+ x
√1−x
We have not derived the derivatives for these functions (yet), but if you are handed the derivative pattern then you should be able to use that pattern to compute derivatives of associated composite functions. 2
arctan ⁡( ( x ) )
74). d
¿
dx Section 2.5: #26, #44
In problems 1–27, differentiate the given function.
26. x ∙ ln(x) – x Problems 43–48 involve parametric equations.
44. Let x(t) = t + 1 and y(t) = t 2 .
(a) Graph (x(t), y(t)) for −1 ≤ t ≤ 4.
dx dy
dy
(b). Find , , thetangent slope ,∧speed whent=1∧t=4
dt dt
dx Section 2.6: #4, #6
4. A young woman and her boyfriend plan to elope, but she must rescue him from his mother who has locked him in his room. The young woman has placed a 20­foot long ladder against his house and is knocking on his window when his mother begins pulling the bottom of the ladder away from the house at a rate of 3 feet per second (see figure below). How fast is the top of the ladder (and the young couple) falling when the bottom of the ladder is: (a) 12 feet from the bottom of the wall? (b) 16 feet from the bottom of the wall? (c) 19 feet from the bottom of the wall? 6. A circle of radius 3 inches is inside a square with 12­inch sides (see figure below). How fast is the area between the circle and square changing if the radius is increasing at 4 inches per minute and the sides are increasing at 2 inches per minute? Section 2.9: #2, #16, #38, #42
dy in two ways: (a) by differentiating implicitly and (b) dx
dy
by explicitly solving for y and then differentiating. Then find the value of at dx
the given point using your results from both the implicit and the explicit differentiation.
2. x 2 + 5y 2 = 45, point: (5, 2)
In Problems 1–10 find dy
using implicit differentiation and then find the slope
dx
of the line tangent to the graph of the equation at the given point.
16. y 2 − 5xy + x 2 + 21 = 0, point: (2, 5)
In Problems 15–22 , find dy
in two ways: (a) by using the “usual” differentiation patterns dx
and (b) by using logarithmic differentiation.
38. y = cos7(2x + 5)
In 32–40, find In 41–46, use logarithmic differentiation to find
42. y = (cos(x)) x dy
dx

 


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