derivative of under root cotx by first principle

Dear student
Let f(x)=cotx.Then f(x+h)=cot(x+h)ddx(f(x))=limh0f(x+h)-f(x)hddx(f(x))=limh0cot(x+h)-cotxhddx(f(x))=limh0cot(x+h)-cotxcot(x+h)+cotxhcot(x+h)+cotxddx(f(x))=limh0cot(x+h)-cotxhcot(x+h)+cotxddx(f(x))=limh0-sinx+h-xsin(x+h) sinxhcot(x+h)+cotx   cotA-cotB=-sin(A-B)sinAsinBddx(f(x))=limh0-sinhsin(x+h) sinxhcot(x+h)+cotxddx(f(x))=limh0-sinhhcot(x+h)+cotxsin(x+h) sinxddx(f(x))=-limh0sinhhlimh01cot(x+h)+cotxsin(x+h) sinxddx(f(x))=-1×12cotxsin2xddx(f(x))=-cosec2x2cotx
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