Could someone please check my work on the following problems? I want to be sure that I'm doing them correctly; i'm just required to find the derivatives and then simplify.

Question

g(t)= t/sint
g'(t)=1*sint-cost*t/(sint)^2
g'(t)=sint-cost*t/(sint)^2

y=ln(e^ax)
y'=1/e^ax*e^x*a

and i'm sort of stuck on this one
y=e^2x[sin(3x)^2]^2
Y'=2^ex*sin(3X)^2+e^2x[2cos3x]
i got this far, but i'm not sure how to simplify that.

Answer 1

my version:
g(t)= t/sint
g'(t) = (sin t -t cos t)/sin^2t
your version:
g'(t)=1*sint-cost*t/(sint)^2
g'(t)=sint-cost*t/(sint)^2

y=ln(e^ax)

my version:
y = a x ln (e) = a x
y' = a

your version:
y'=1/e^ax*e^x*a times a because d/dx(e^ax) = a e^ax

but that is just x a

and i'm sort of stuck on this one
y=e^2x[sin(3x)^2]^2

my version:
[sin(3x)^2]^2 * 2 e^2x + e^2x * 2 [sin(3x)^2]^1 * d/dx(sin(3x)^2)

= 2 e^2x *{[sin(3x)^2]^2 + sin(3x)^2 * sin(3x)^2* [cos(3x)^2] *2*3x*3]

= 2 e^2x * sin(3x)^2]^2 *{ 1 + 18xcos(3x)^2}
wow - check my arithmetic

Your version:

Y'=2^ex*sin(3X)^2+e^2x[2cos3x]