Question

If y=e sin(cosec-1x )then dy upon dx​

Answer 1

Given : The function is $y=e^{sin(cosec^{-1}x)}$

To find : first derivative of given function i.e., dy/dx

solution : $y=e^{sin(cosec^{-1}x)}$

differentiating with respect to x,

⇒dy/dx = $\frac{d(e^{sin(cosec^{-1}x)})}{dx}$

= $e^{sin(cosec^{-1}x)}\times\frac{d(sin(cosec^{-1}x))}{dx}$

= $e^{sin(cosec^{-1}x)} \times cos(cosec^{-1}x)\times\frac{d(cosec^{-1}x}{dx}$

= $e^{sin(cosec^{-1}x)} \times cos(cosec^{-1}x)\times\frac{-1}{x\sqrt{x^2-1}}$

=$-e^{sin(cosec^{-1}x)} \times cos(cosec^{-1}x)\times\frac{1}{x\sqrt{x^2-1}}$

but sin(cosec¯¹x) = 1/x

so, $e^{sin(cosec^{-1}x)}=\sqrt[x]{e}$

cos(cosec¯¹x) = √(x² - 1)/x

Then $-e^{sin(cosec^{-1}x)} \times cos(cosec^{-1}x)\times\frac{1}{x\sqrt{x^2-1}}=-\sqrt[x]{e}\times\frac{\sqrt{x^2-1}}{x}\times\frac{1}{x\sqrt{x^2-1}}$

= $-\frac{\sqrt[x]{e}}{x^2}$

Therefore the first derivative of function y is $-\frac{\sqrt[x]{e}}{x^2}$

Answer 2

Answer: Let's make it simple

sin(cosec^-1x)=1/x

y=e^(1/x)

dy/dx= -e^(1/x)/x² { on complete first order differentiation}

Hope it helps

best luck buddy