How do you find the derivative of #arcsin(5x)#?

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Answer 1 · 2018-04-17T09:59:50

#y'=5/sqrt(1-(5x)^2)#

#y=sin^-1(5x)#

Substitute for #x#

#sinu=5x#

#cosu*du=5dx#

#(du)/dx=5/cosu# #color(green)(rarr(1)#

After substitution,

#y=sin^-1sinu#

#color(blue)(sin^-1sinu=u#

#y=u#

Differentiate with respect to #x#

#y'=(du)/dx#

Substitute from #color(green) ((1)#

#y'=5/cosu#

#y'=5/sqrt(1-sin^2u)#

Reverse the substitution

#y'=5/sqrt(1-(5x)^2)#

#color(red) "and the general formula to find the derivative of arcsin functions"#

#color(green)(d/dxsin^-1u=1/sqrt(1-u^2)(du)/dx#