Question #16796

The function f(x) = sin(x) with domain D = [-pie/2 ; pie/2 ] is one-to-one. Its inverse function is called f^-1(y) = arcsin(y). (a) Determine the domain and range of arcsin. (b) Calculate the derivative d/dy arctan(y).[Use implicit differentiation]

Expert's answer

Domain is (-1,1)

Range is (-pi/2,pi/2)

x=arctgy

y=tgx

dy/dx=1/cos^2(x)

dx/dy=cos^2(x)

1/cos^2x=1+y^2

cos^2=1/(1+y^2)

x'=1/(1+y^2)

Range is (-pi/2,pi/2)

x=arctgy

y=tgx

dy/dx=1/cos^2(x)

dx/dy=cos^2(x)

1/cos^2x=1+y^2

cos^2=1/(1+y^2)

x'=1/(1+y^2)

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