Clearly I'm missing at least one important step.

(1) Find an equation of the line tangent to the curvey1 given below that is parallel to the given liney2.

1=6x * sqrt(x)

y

y2 = 9 + 9x

y1 = 6x(x^.5)

y1 = 6x^(3/2)

y1' = (3/2)(6x^.5)

y1' = 9x^(1/2)

Seems to match the criteria, but it's wrong?

(2) For what values ofrdoes the functiony=erxsatisfy the equationy'' + 3y' - 40y= 0?

(3)f(x) = 9 cos(x) + 5/7 cot(x)

Seemed simple.

9(-sin(x)) + (5/7)(-csc^2(x))

(4) 6-6tan(x) /sin(x)-cos(x)

limit as x goes to pi/4. Bottom is 0 = ??

(5) Implicit differentiation cos(x) + sqrt(y) = 7

-sin(x) + (1/2)y^(-1/2) * (dy/dx) = 0

(1/2)y^(-1/2) = sin(x)

y^(-1/2) = 2sin(x)

1/sqrt(y) = 2sin(x)

y = (1/2sin(x))^2

y = 1/4sin(x)^2

Again, not sure where I went wrong.

(6) 5sqrt(x) + sqrt(y) = 5

(5/2)x^(-1/2) + (1/2)y^(-1/2) * (dy/dx) = 0

I can't put this answer into a form that doesn't suck, but it was 5/sqrt(x) over 1/sqrt(y)

(7) f(x) = sin(4x) + ln(4x)

f'(x) = cos(4x) + 1/4x

This one seems simple, but I'm guessing the problem is with the Trig part.