Question

Just a deal of gross multiplication regarding implicit differentiation and trigonometric functions (Yeah, i'm bad at basic math, lol. Question below.)

Answer 1

I'm finding the derivative of \[3^{\sin (\theta)}\]By taking the natural log of both sides (assuming that is a y is a function of x) and getting this:\[y = 3^{\sin(\theta)}\]\[\ln(y) = \ln(3^{\sin (\theta)}) = \sin (\theta) \ln(3)\]\[\frac{ 1 }{ y }\frac{ dy }{ dx }=\ln(3)\cos(\theta) + \sin(\theta)(\frac{ 1 }{ 3 })\]\[\frac{ dy }{ dx } = [\ln(3)\cos(\theta) +
\sin(\theta)(\frac{ 1 }{ 3 })]*[3^{\sin(\theta)}]\][substituting y for 3^sin(theta)]

Now, the answer should be \[3^{\sin (\theta)}\ln(3)\cos(\theta)\]Which means that the second term has to cancel entirely. I'm just horrible at some components of basic math when it comes to trig values in exponents, how do I deal with that second term being multiplied?