Troubleshooting Differentiation: What Went Wrong with (1 - sin x) / (1 + sin x)?

[maccaman]

Differentiate y = (1 - sin x) / (1 + sin x)

i know the answer is - 2 cos x / 1 + sin x

Again, i just can't do the working

 

[Zorodius]

Your answer is wrong. The denominator should be squared.

You can either use the product and chain rule on the equivalent expression (1 - sin x)(1 + sin x)^-1, or you can use the quotient rule.

The latter is more straightforward, so let's go with that one.

The quotient rule is:

$$\frac {d}{dx} \left[ \frac {f(x)}{g(x)} \right] = \frac {g(x)f'(x) - f(x)g'(x)}{g(x)^2}$$

Be careful with that thing, it's dangerous! It's sharp and pointy and you shouldn't run while holding it. And it matters which function is on the top, so make sure you know which is f(x) and which is g(x) before using it.

Applying the quotient rule yields:

$$\frac {d}{dx} \frac {1 - \sin x}{1 + \sin x} = \frac {(1 + \sin x)(-\cos x) - (1 - \sin x)(\cos x)}{(1 + \sin x)^2}$$

Factor out -cos x:

$$= -\cos x \left[ \frac{1 + \sin x + 1 - \sin x}{(1 + \sin x)^2} \right]$$

Simplify:

$$= \frac{-2 \cos x}{(1 + \sin x)^2}$$

 

[maccaman]

god dammit, stupid textbook
yeh thanks for that, i got that answer too, i was wondering where i went wrong as i got (1 + sin x) squared on the bottom too, but the text got 1 + sin x on the bottom, arggh. thanks for that