I'm reviewing and studying for finals and I've run into some trouble with the chain rule.
I have no problem using the chain rule to differentiate problems like this:
y = (3x + 1)^2
y'= 2(3x + 1)(3) = 6(3x +1)
y = 2^cotx
lny = ln(2^cotx) = cotx*ln2
y'/y = cotx(1/2*0) + ln2*-csc^2x
But when I comes across problems like this, using the chain rule becomes gets confusing:
y = 2^sin3.14x
I don't know how to differentiate sin3.14x because I can't distinguish between the first and second term. Is the first term sin3.14 and the second x, is the first term sin and the second 3.14x, or should it all be treated as one term?
Anyway, this how I have tried to slove it:
lny = ln(2^sin3.14x) = sin3.14x*ln2
y'/y = sin3.14x(1/2*0) + ln2(sin3.14 + cos3.14x)
y' = y(ln2(sin3.14 + cos3.14x)
y' = 2^sin3.14x(ln2(sin3.14 + cos3.14x))
It's supposed to be y' = 2^sin3.14x(3.14ln2)cos3.14x
Will someone kindly tell me what I'm doing wrong?