$h(x) = \frac{1}{f(x)}$

cosec x = \frac{1}{s i n x} cosec^{2} x = \frac{1}{s i n ^{2} x} sec x = \frac{1}{c o s x} cot x = \frac{1}{t a n x}

Rules for differentiating $\frac{1}{f(x)}$

$f(x)$	$f'(x)$
$\tan kx$	$k\sec^{2}kx$
$\sec kx$	$k \sec kx \times \tan kx$
$\cot kx$	$-k \space \text{cosec}^2 \space kx$
$\text{cosec } kx$	$-k \space \text{cosec} \space kx \times \cot kx$

y = tan x y = \frac{s i n x}{c o s x} u = sin x v = cos x \frac{d y}{d x} = cos x \frac{d u}{d x} = - sin x \frac{d y}{d x} = \frac{v ( x ) u ^{'} ( x ) - u ( x ) v ^{'} ( x )}{v ( x ) ^{2}} \frac{d y}{d x} = \frac{c o s x \times c o s x -- s i n x \times s i n x}{( c o s x ) ^{2}} \frac{d y}{d x} = \frac{c o s ^{2} x + s i n ^{2} x}{c o s ^{2} x} \frac{d y}{d x} = \frac{1}{c o s ^{2} x} \frac{d y}{d x} = sec^{2} x

y = cosec x y = \frac{1}{s i n x} y = (sin x)^{- 1} y = u^{- 1} u = sin x \frac{d y}{d u} = - u^{- 2} \frac{d y}{d u} = \frac{- 1}{u ^{2}} \frac{d u}{d x} = cos x \frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x} \frac{d y}{d x} = cos x \times \frac{- 1}{s i n ^{2} x} \frac{d y}{d x} = \frac{c o s x}{s i n x} \times \frac{- 1}{s i n x} \frac{d y}{d x} = cot x \times - cosec x \frac{d y}{d x} = - cot x \times cosec x

Todo

arcsin x = sin^{- 1} x arccos x = cos^{- 1} x arctan x = tan^{- 1} x