Question

find the derivative of tanx by 1st principal

Answer 1

general, the derivative of any function say f(x)f(x) is given as

f′(x)=limh→0f(x+h)−f(x)hf′(x)=limh→0f(x+h)−f(x)h

∴ddx(tanx)=limh→0tan(x+h)−tan(x)h∴ddx(tan⁡x)=limh→0tan⁡(x+h)−tan⁡(x)h

=limh→0sin(x+h)cos(x+h)−sinxcosxh=limh→0sin⁡(x+h)cos⁡(x+h)−sin⁡xcos⁡xh

=limh→0sin(x+h)cosx−cos(x+h)sinxhcosxcos(x+h)=limh→0sin⁡(x+h)cos⁡x−cos⁡(x+h)sin⁡xhcos⁡xcos⁡(x+h)

=limh→0sin(x+h−x)hcosxcos(x+h)=limh→0sin⁡(x+h−x)hcos⁡xcos⁡(x+h)

=limh→0sinhhcosxcos(x+h)=limh→0sin⁡hhcos⁡xcos⁡(x+h)

=limh→0sinhh⋅limh→01cosxcos(x+h)=limh→0sin⁡hh⋅limh→01cos⁡xcos⁡(x+h)

=1⋅1cosx⋅cosx=1⋅1cos⁡x⋅cos⁡x

=1cos2x=1cos2⁡x

=sec2x