y = ${\mathit{x}}^{\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ and z = $\frac{{\mathit{x}}^2+1}{{\mathit{x}}^2-1}$
Consider y = ${\mathit{x}}^{\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$
Taking log on both sides,
log y = log $\left({\mathit{x}}^{\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}\right)$
log y = x cos x log x
Differentiating with respect to x,
$\frac{1}{\mathit{y}}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = (x cos x) $\frac{1}{\mathit{x}}$ + log x $\frac{\mathit{d}}{\mathit{d}\mathit{x}}$ + log x $\frac{\mathit{d}}{\mathit{d}\mathit{x}}$ (x cos x)
$\frac{1}{\mathit{y}}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = cos x + log x (cos x - x sin x)
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = y (cos x + log x [cos x - x sin x)]
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = ${\mathit{x}}^{\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ [cos x + log x (cos x - x sin x)] … (1)
Consider z = $\frac{{\mathit{x}}^2+1}{{\mathit{x}}^2-1}$
Differentiating with respect to x,
$\frac{\mathit{d}\mathit{z}}{\mathit{d}\mathit{x}}$ =
= $\frac{\left({\mathit{x}}^2-1\right)\left(2\mathit{x}\right)-\left({\mathit{x}}^2+1\right)\left(2\mathit{x}\right)}{{\left({\mathit{x}}^2-1\right)}^2}$
= $\frac{2{\mathit{x}}^3-2\mathit{x}-2{\mathit{x}}^3-2\mathit{x}}{{\left({\mathit{x}}^2-1\right)}^2}$
= $\frac{-4\mathit{x}}{{\left({\mathit{x}}^2-1\right)}^2}$ ... (2)
Adding (1) and (2):
$\frac{\mathit{d}}{\mathit{d}\mathit{x}}\left\{{\mathit{x}}^{\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}+\frac{{\mathit{x}}^2+1}{{\mathit{x}}^2-1}\right\}$ = $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}+\frac{\mathit{d}\mathit{z}}{\mathit{d}\mathit{x}}$
= ${\mathit{x}}^{\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ [cos x + log x (cos x – x sin x)] – $\frac{\mathit{x}}{{\left({\mathit{x}}^2-1\right)}^2}$
OR
x = a(θ - sinθ) , y = a(1 + cosθ)
Differentiating x and y w.r.t. θ,
$\frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{\theta }}$ = a (1 - cos θ) ... (1)
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{\theta }}$ = - a sin θ ... (2)
Dividing (2) by (1),
$\frac{\left(\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{\theta }}\right)}{\left(\frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{\theta }}\right)}$ = $\frac{-\mathit{a}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }}{\mathit{a}\left(1-\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\right)}$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{-\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }}{1-\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }}$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{-2\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{\theta }}{2}\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{\theta }}{2}}{2\mathit{s}\mathit{i}{\mathit{n}}^2\frac{\mathit{\theta }}{2}}$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{-\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{\theta }}{2}}{\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{\theta }}{2}}$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = = - cot $\frac{\mathit{\theta }}{2}$ ... (3)
Differentiating w.r.t. x,
$\frac{\mathit{d}}{\mathit{d}\mathit{x}}\left(\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right)$ = $\frac{\mathit{d}}{\mathit{d}\mathit{\theta }}\left(\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right)\times \frac{\mathit{d}\mathit{\theta }}{\mathit{d}\mathit{x}}$
⇒ $\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$ = $\frac{\mathit{d}}{\mathit{d}\mathit{\theta }}\left(\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right)\times \frac{\mathit{d}\mathit{\theta }}{\mathit{d}\mathit{x}}$
⇒ $\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$ = $\frac{\mathit{d}}{\mathit{d}\mathit{\theta }}\left(-\mathit{c}\mathit{o}\mathit{t}\frac{\mathit{\theta }}{2}\right)\times \frac{\mathit{d}\mathit{\theta }}{\mathit{d}\mathit{x}}$ [from equation (3)]
$\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$ = - $\left(-\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\frac{\mathit{\theta }}{2}\times \frac{1}{2}\right)\times \frac{\mathit{d}\mathit{\theta }}{\mathit{d}\mathit{x}}$
= $\frac{1}{2}\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\frac{\mathit{\theta }}{2}\times \frac{1}{\left(\frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{\theta }}\right)}$
= $\frac{1}{2}\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\frac{\mathit{\theta }}{2}\times \frac{1}{\mathit{a}\left(1-\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\right)}$ ... [from equation (1)]
= $\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\frac{\mathit{\theta }}{2}}{2\mathit{a}\left(1-\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\right)}$
= $\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\frac{\mathit{\theta }}{2}}{2\mathit{a}\left(2\mathit{s}\mathit{i}{\mathit{n}}^2\frac{\mathit{\theta }}{2}\right)}$
= $\frac{1}{4\mathit{a}}\times \mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^2\frac{\mathit{\theta }}{2}$