( 1 ) g ( x ) = 1 s e c − 1 ( 4 x ) + c s c − 1 ( 4 x ) Differentiating wrt x on both sides, g ′ ( x ) = − 1 ( s e c − 1 ( 4 x ) + c s c − 1 ( 4 x ) ) 2 × d d x ( s e c − 1 ( 4 x ) + c s c − 1 ( 4 x ) ) = − 1 ( s e c − 1 ( 4 x ) + c s c − 1 ( 4 x ) ) 2 × ( x 2 4 16 − x 2 ( − 4 x 2 ) − 1 4 ∣ x ∣ 16 x 2 − 1 ) = 1 ( s e c − 1 ( 4 x ) + c s c − 1 ( 4 x ) ) 2 × ( 1 16 − x 2 + 1 4 ∣ x ∣ 16 x 2 − 1 ) ( 2 ) (1)\\g(x)=\dfrac{1}{sec^{-1}(\frac{4}{x})+csc^{-1}(4x)}\\
\text{Differentiating wrt x on both sides,}\\
g'(x)\\=-\dfrac{1}{(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))^2}\times \dfrac{d}{dx}(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))\\
=-\dfrac{1}{(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))^2}\times \left(\dfrac{x^2}{4\sqrt{16-x^2}}\left(-\dfrac{4}{x^2}\right)-\dfrac{1}{4|x|\sqrt{16x^2-1}{}}\right)\\\;\\
=
\boxed{\dfrac{1}{(sec^{-1}(\frac{4}{x})+csc^{-1}(4x))^2}\times \left(\dfrac{1}{\sqrt{16-x^2}}+\dfrac{1}{4|x|\sqrt{16x^2-1}{}}\right)}\\
\;\\
(2) ( 1 ) g ( x ) = se c − 1 ( x 4 ) + cs c − 1 ( 4 x ) 1 Differentiating wrt x on both sides, g ′ ( x ) = − ( se c − 1 ( x 4 ) + cs c − 1 ( 4 x ) ) 2 1 × d x d ( se c − 1 ( x 4 ) + cs c − 1 ( 4 x )) = − ( se c − 1 ( x 4 ) + cs c − 1 ( 4 x ) ) 2 1 × ( 4 16 − x 2 x 2 ( − x 2 4 ) − 4∣ x ∣ 16 x 2 − 1 1 ) = ( se c − 1 ( x 4 ) + cs c − 1 ( 4 x ) ) 2 1 × ( 16 − x 2 1 + 4∣ x ∣ 16 x 2 − 1 1 ) ( 2 )
f ( x ) = 6 s i n − 1 1 − x 2 Differentiating wrt x on both sides, f ′ ( x ) = 6 1 − ( 1 − x 2 ) 2 × d d x ( 1 − x 2 ) = 6 ∣ x ∣ × − 2 x 2 1 − x 2 = − 6 x ∣ x ∣ 1 − x 2 ( 3 ) f(x)=6sin^{-1}\sqrt{1-x^2}\\
\text{Differentiating wrt x on both sides,}\\
f'(x)\\
=\dfrac{6}{\sqrt{1-\left(\sqrt{1-x^2}\right)^2}}\times\dfrac{d}{dx}\left(\sqrt{1-x^2}\right)\\
=\dfrac{6}{|x|}\times \dfrac{-2x}{2\sqrt{1-x^2}}\\\;\\
=\boxed{-\dfrac{6x}{|x|\sqrt{1-x^2}}}\\\;\\
(3) f ( x ) = 6 s i n − 1 1 − x 2 Differentiating wrt x on both sides, f ′ ( x ) = 1 − ( 1 − x 2 ) 2 6 × d x d ( 1 − x 2 ) = ∣ x ∣ 6 × 2 1 − x 2 − 2 x = − ∣ x ∣ 1 − x 2 6 x ( 3 )
h ( x ) = x s i n − 1 ( 2 x ) Differentiating wrt x on both sides, h ′ ( x ) = x 1 − 4 x 2 × 2 + s i n − 1 ( 2 x ) = 2 x 1 − 4 x 2 + s i n − 1 ( 2 x ) ( 4 ) h(x)=xsin^{-1}(2x)\\
\text{Differentiating wrt x on both sides,}\\
h'(x)\\
=\dfrac{x}{\sqrt{1-4x^2}}\times2+sin^{-1}(2x)\\\;\\
=\boxed{\dfrac{2x}{\sqrt{1-4x^2}}+sin^{-1}(2x)}
\\\;\\
(4) h ( x ) = x s i n − 1 ( 2 x ) Differentiating wrt x on both sides, h ′ ( x ) = 1 − 4 x 2 x × 2 + s i n − 1 ( 2 x ) = 1 − 4 x 2 2 x + s i n − 1 ( 2 x ) ( 4 )
y = s e c − 1 x 2 Differentiating wrt x on both sides, d y d x = 1 ∣ x ∣ x 2 − 1 × 2 x 2 x 2 = x x 2 x 2 − 1 = 1 x x 2 − 1 y=sec^{-1}\sqrt{x^2}\\
\text{Differentiating wrt x on both sides,}\\
\dfrac{dy}{dx}\\
=\dfrac{1}{|x|\sqrt{x^2-1}}\times \dfrac{2x}{2\sqrt{x^2}}\\\;\\
=\dfrac{x}{x^2\sqrt{x^2-1}}=\boxed{\dfrac{1}{x\sqrt{x^2-1}}} y = se c − 1 x 2 Differentiating wrt x on both sides, d x d y = ∣ x ∣ x 2 − 1 1 × 2 x 2 2 x = x 2 x 2 − 1 x = x x 2 − 1 1
( 5 ) y = c o s − 1 ( s i n x ) Differentiating wrt x on both sides, d y d x = − 1 1 − s i n 2 x × c o s x = − c o s x ∣ c o s x ∣ (5)\\
y=cos^{-1}(sinx)\\
\text{Differentiating wrt x on both sides,}
\\
\dfrac{dy}{dx}\\
=
-\dfrac{1}{\sqrt{1-sin^2x}}\times cosx\\=\boxed{-\dfrac{cosx}{|cosx|}} ( 5 ) y = co s − 1 ( s in x ) Differentiating wrt x on both sides, d x d y = − 1 − s i n 2 x 1 × cos x = − ∣ cos x ∣ cos x
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