A)
Y = x + 1 x Y=\sqrt{x}+\dfrac{1}{\sqrt{x}} Y = x + x 1
d y d x = 1 2 x + 1 2 x 3 \dfrac{dy}{dx}=\dfrac{1}{2\sqrt x}+\dfrac{1}{2\sqrt[3]{x}} d x d y = 2 x 1 + 2 3 x 1
B)
Y = x 2 + π 2 + x π Y=\ x^{2}+\pi^{2}+x^{\pi} Y = x 2 + π 2 + x π
d y d x = 2 x + π x π − 1 \dfrac{dy}{dx}=2x+\pi{x}^{\pi-1} d x d y = 2 x + π x π − 1
C)
Y = s i n x c o s x − 1 Y=\dfrac{sinx}{cosx-1} Y = cos x − 1 s in x
d y d x = c o s x × c o s x − ( s i n x − 1 ) ( − s i n x ) ( c o s x − 1 ) 2 \dfrac{dy}{dx}=\dfrac{cosx\times cosx-(sinx-1)(-sinx)}{(cosx-1)^2} d x d y = ( cos x − 1 ) 2 cos x × cos x − ( s in x − 1 ) ( − s in x )
d y d x = c o s 2 x + s i n 2 x − s i n x ( c o s x − 1 ) 2 \dfrac{dy}{dx}=\dfrac{cos^{2}x+sin^{2}x-sinx}{{{(cosx-1}})^{2}} d x d y = ( cos x − 1 ) 2 co s 2 x + s i n 2 x − s in x
d y d x = 1 − s i n x ( c o s x − 1 ) 2 \dfrac{dy}{dx}=\dfrac{1-sinx}{(cosx-1)^{2}} d x d y = ( cos x − 1 ) 2 1 − s in x
D)
Y = x 2 s e c x Y=x^{2}secx Y = x 2 sec x
d y d x = 2 × x × s e c x + s e c x × t a n x × x 2 \dfrac{dy}{dx}=2\times x\times secx+secx\times tanx \times x^{2} d x d y = 2 × x × sec x + sec x × t an x × x 2
=xsecx(xtanx+2)
E)
Y = 1 e x + 2 Y=\dfrac{1}{e^{x}+2} Y = e x + 2 1
d y d x = e x + 2 × 0 + 1 × e x ( e x + 2 ) 2 = e x ( e x + 2 ) 2 \dfrac{dy}{dx}=\dfrac{e^{x}+2\times0+1\times e^{x}}{(e^x+2)^2}=\dfrac{e^{x}}{(e^x+2)^2} d x d y = ( e x + 2 ) 2 e x + 2 × 0 + 1 × e x = ( e x + 2 ) 2 e x
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