Question

From first principle find the derivative of ln(cosec x^2)​

Answer 1

Answer:

f(x)=cscx

f(x+h)=csc(x+h)

f(x+h)−f(x)=csc(x+h)−cscx=1sin(x+h)−1sinx

=sinx−sin(x+h)sin(x+h)(sinx)

=2sin(x−x−h2)cos(x+x+h2)sin(x+h)(sinx)

=−2sin(h2)cos(2x+h2)sin(x+h)(sinx)

f′(x)=limh→0f(x+h)−f(x)h

=limh→0−2sin(h2)cos(2x+h2)hsin(x+h)(sinx)

=−limh→0sin(h2)h2×limh→0cos(2x+h2)sin(x+h)sinx

=−cosxsin2x

=−cscxcotx

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