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Urgent! Please help me! I have an exam in two days and I really need help with these questions. Please explain the steps and answers as thoroughly as possible.

1. Find the derivative f'(x).
(a) f(x)=xexcosx
(b) f(x)=secxtanx
(c) f(x)=sinx/1+cosx
 
2. Let g(x) be a differentiable function such that g(0)=2 and g'(0)=3.
(a) f(x)=(g(x))3
(b) f(x)=g(7x)
 
3. If h(t)=300+12sin(t/2), find h"(pi/3).
 
4. Find the derivative f'(x).
(a) f(x)=(4-2x-3x2)-5
(b) f(x)=x(square root of 1+x2)
(c) f(x)=cos2(x2)
 
5. Find the derivative f'(x).
(a) f(x)=lnx/x
(b) f(x)=arctan(ex)
(c) f(x)=xlnx
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1 Answer

I'll do part (a) of each problem and give somebody else the chance to do the rest. :)
 
1a. Use the product rule, (uv)'=u'v+uv'.
For a product of 3 functions, (uvw)'=u'vw+uv'w+uvw'
(xexcosx)'=excosx+xexcosx-xexsinx
 
2a. Use the chain rule, f(g(x))'=f'(g(x)) g'(x).
(g(x)³)'=3g(x)² g'(x)
(g³)'(0)=3g(0)² g'(0)=3*2²*3=36
 
3. h(t)=300+12sin(t/2)
h'(t)=6cos(t/2)
h''(t)=-3sin(t/2)
h''(Pi/3)=-3sin(Pi/6)=-3/2
 
4a. Chain rule again: outside derivative * inside derivative
f(x)=(4-2x-3x²)-5
f'(x)=-5(4-2x-3x²)-6*(-2-6x)
 
 
5a. Quotient rule or product rule, I'll use the product rule:
 
(ln(x)/x)'=(x-1*ln(x))' =-x-2ln(x)+x-1*x-1=-x-2ln(x)+x-2
 
Good luck on your exam! :)

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