Derivative of $\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\tan \frac x2\right)$.

Question

Find the derivative of $\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\tan \frac x2\right)$.

I'm learning differentiation and this is an exercise problem from my book. I used chain rule and got the following:

$\begin{align} \dfrac d{dx}\left[\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\tan\frac x2\right)\right] &= \dfrac{1}{1+\frac{a-b}{a+b}\tan^2\frac x 2}\cdot\dfrac{d}{dx}\left(\sqrt{\frac{a-b}{a+b}}\tan \frac x2\right)\\ &= \dfrac{1}{1+\frac{a-b}{a+b}\tan^2\frac x 2}\cdot\frac 1 2\sqrt{\frac{a-b}{a+b}}\sec^2\frac x 2 \end{align}$

But this doesn't match the answer in the book. The given answer is $\frac{\sqrt{a^2-b^2}}{2(a+b\cos x)}$. So, where did I go wrong and what is the correct solution?

Answer 1

Power of $t$-formula

Let $t=\tan \dfrac{x}{2}$, then

\begin{align} \frac{dt}{dx} &= \frac{1}{2} \sec^2 \frac{x}{2} \\ &= \frac{1+t^2}{2} \\ \tan y &= t\sqrt{\frac{a-b}{a+b}} \\ \sec^2 y \times \frac{dy}{dx} &= \frac{dt}{dx} \times \sqrt{\frac{a-b}{a+b}} \\ \left( 1+\frac{a-b}{a+b} t^2 \right) \frac{dy}{dx} &= \frac{1+t^2}{2} \times \sqrt{\frac{a-b}{a+b}} \\ \frac{dy}{dx} &= \sqrt{\frac{a-b}{a+b}} \times \frac{(a+b)(1+t^2)}{2[a+b+(a-b)t^2]} \\ &= \frac{\sqrt{a^2-b^2}}{2} \times \frac{1+t^2}{a(1+t^2)+b(1-t^2)} \\ &= \frac{\frac{1}{2} \sqrt{a^2-b^2}} {a+b\left( \frac{1-t^2}{1+t^2} \right)} \\ &= \frac{\sqrt{a^2-b^2}}{2(a+b\cos x)} \end{align}

Answer 2

Just keep going from where you stopped: Taking things one or two steps at a time, we have

$$\begin{align} {1\over1+{a-b\over a+b}\tan^2{x\over2}}\cdot{1\over2}\sqrt{a-b\over a+b}\sec^2{x\over2} &={a+b\over a+b+(a-b)\tan^2{x\over2}}\cdot{1\over2}\sqrt{a-b\over a+b}\sec^2{x\over2}\\ &={1\over2}\cdot{\sqrt{(a+b)(a-b)}\over a+b+(a-b)\tan^2{x\over2}}\cdot{1\over\cos^2{x\over2}}\\ &={1\over2}\cdot{\sqrt{a^2-b^2}\over(a+b)\cos^2{x\over2}+(a-b)\sin^2{x\over2}}\\ &={1\over2}\cdot{\sqrt{a^2-b^2}\over a\left(\cos^2{x\over2}+\sin^2{x\over2}\right)+b\left(\cos^2{x\over2}-\sin^2{x\over2}\right)}\\ &={1\over2}\cdot{\sqrt{a^2-b^2}\over a+b\cos x} \end{align}$$