Question

Derive (sinx)/(1+tanx)

Answer 1

You use the quotient rule for derivatives since you are dividing.
We will use F(x) for sinx and G(x) for (1+tanx), and H'(x) for the final answer.
So H'(x) = (F'(x)G(x) - F(x)G'(x))/(G(x))^(2))
The derivative of sinx is cosx so F'(x) = cosx
The derivative of 1+tanx is sec^(2)x, just like the derivative of tangent (because the derivative of a constant is 0, and by the addition rule for derivatives you would add the derivatives). 0 + sec^(2)x is sec^(2)x. So G'(x) is sec^(2)x.
So with this information just plug F, F', G, and G' into the H'(x) = (F'(x)G(x) - F(x)G'(x))/(G(x))^(2)) equation and solve.
H'(x) = (cosx)(1+tanx) - (sinx)(sec^(2)x)/((1+tanx)^(2))