What is the derivative of sin(a/2 + d/2)/sin(a/2)?

Question

I'm embarrassed to say that I just can't seem to come up with the correct answer. I get a different answer if I use the product method rather than the quotient method. This should be a super easy one.

Thanks in advance for your help!

Greg

Answer 1

With respect to what variable would you like to find the derivative? I'm going to assume that it's respect to a, because the derivative is more challenging that way.

f(a) = sin(a/2 + d/2)/sin(a/2)

The direct method is to use the quotient rule.

f'(a) = [cos(a/2 + d/2)(1/2)sin(a/2) - sin(a/2 + d/2)cos(a/2)(1/2)] / sin^2(a/2)

f'(a) = [(1/2)sin(a/2)cos(a/2 + d/2) - (1/2)cos(a/2)sin(a/2 + d/2)] / sin^2(a/2)

Answer 2

what is the independent variable here? The main thing you need to do is simplify the expression before taking derivatives: use the sum of angles formula for the sin function:

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

Answer 3

Assuming d to be constant with respect to a,
f(a) = sin(a/2 + d/2) / sin(a/2)
f'(a) = (1/2)(sin(a/2)cos(a/2 + d/2) - cos(a/2)sin(a/2 + d/2)) / sin^2(a/2)
sin (s - t) = sin s cos t - cos s sin t
f'(a) = (1/2)sin(a/2 - a/2 - d/2) / sin^2(a/2)
f'(a) = -(1/2)sin(d/2) / sin^2(a/2)