I think I have the correct answers for the following problems. For anyone who has the time I would appreciate it if you could tell me if I am correct/incorrect. Thank you.

1). Differentiate �ã(x)sin x = (2xcosx+sinx)/2�ã(x)

2). Differentiate 2t/(4+t^2) = (8-2t^2)/(4+t^2)^2

3). Differentiate (1-secx)/(tanx) = secx(-tan^2x + sec^2x)/(tanx)^2

4). Find equation of tangent line
y=(1+x)(cos x)

tangent line = y= x+1

5). If f(3)=4 g(3)=2 f'(3)=-6 g'(3)=5
A).(f+g)(3) = 6
B).(fg)(3) = 8
C).(f/g)(3)= -8

Note: �ã(x) is supposed to be square root of x.

Sure, I'd be happy to help you check your answers!

1) To differentiate the function �ã(x)sin(x) = (2xcos(x)+sin(x))/2�ã(x), you can use the product rule. The product rule states that if you have a function of the form f(x)g(x), then the derivative of that function is f'(x)g(x) + f(x)g'(x).

In this case, f(x) = �ã(x) and g(x) = sin(x). The derivatives of f(x) and g(x) are f'(x) = 1/(2�ã(x)) and g'(x) = cos(x), respectively. Applying the product rule, we get:

�ã(x)sin(x) = (�ã(x))(1/(2�ã(x)))(sin(x)) + (�ã(x))(cos(x))
= sin(x)/2�ã(x) + �ã(x)cos(x).

So your answer is correct.

2) To differentiate the function 2t/(4+t^2) = (8-2t^2)/(4+t^2)^2, you can use the quotient rule. The quotient rule states that if you have a function of the form f(x)/g(x), then the derivative of that function is (f'(x)g(x) - f(x)g'(x))/(g(x))^2.

In this case, f(x) = 2t and g(x) = 4+t^2. The derivatives of f(x) and g(x) are f'(x) = 2 and g'(x) = 2t, respectively. Applying the quotient rule, we get:

2t/(4+t^2) = (2)(4+t^2) - (2t)(2t)/(4+t^2)^2
= 8 + 2t^2 - 4t^2/(4+t^2)^2
= 8 - 2t^2/(4+t^2)^2.

So your answer is correct.

3) To differentiate the function (1-sec(x))/(tan(x)) = sec(x)(-tan^2(x) + sec^2(x))/(tan(x))^2, you can use the quotient rule as well. The derivatives of sec(x) and tan(x) are sec(x)tan(x) and sec^2(x), respectively. Applying the quotient rule, we get:

(1-sec(x))/(tan(x)) = (tan(x))(sec(x)tan(x) - sec^2(x))/(tan(x))^2
= sec(x)tan(x) - sec^2(x)/(tan(x)).

So your answer is correct.

4) To find the equation of the tangent line to the curve y = (1+x)(cos(x)), we need to find the derivative of the function with respect to x. The derivative of (1+x) is 1, and the derivative of cos(x) is -sin(x). Thus, the derivative of y with respect to x is:

dy/dx = 1*cos(x) + (1+x)*(-sin(x))
= cos(x) - sin(x) - x*sin(x).

To find the equation of the tangent line at a specific point, you need both the derivative and the coordinates of that point. Since you didn't provide a specific point, we cannot verify your answer.

5) To find the values of (f+g)(3), (fg)(3), and (f/g)(3), we need to use the given functions and their derivatives:

f(x) = 4, g(x) = 2
f'(x) = -6, g'(x) = 5.

A) (f+g)(3) = f(3) + g(3) = 4 + 2 = 6. Your answer is correct.

B) (fg)(3) = f(3) * g(3) = 4 * 2 = 8. Your answer is correct.

C) (f/g)(3) = f(3) / g(3) = 4 / 2 = 2. Your answer is incorrect; it should be 2, not -8.

I hope this helps clarify your answers. Let me know if you have any further questions or need additional explanations!