#d/(dx)(tan^(-1).((4x)/sqrt(1-4x^2)))=?#

1 Answer
Dec 22, 2017

#4/((1+12x^2)(1-4x^2)^(1/2))#

Explanation:

#•color(white)(x)d/dx(tan^-1(f(x)))=1/(1+(f(x))^2)xxf'(x)#

#rArrd/dx(tan^-1((4x)/sqrt(1-4x^2)))#

#=1/(1+((4x)/sqrt(1-4x^2))^2)xxd/dx((4x)/sqrt(1-4x^2))#

#=1/(1+(16x^2)/(1-4x^2))xx....#

#=1/((1-4x^2+16x^2)/(1-4x^2))xx....#

#=(1-4x^2)/(1+12x^2)xxd/dx((4x)/sqrt(1-4x^2))#

#"differentiate using "color(blue)"quotient/chain rule"#

#"given "y=(g(x))/(h(x))" then"#

#dy/dx=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#

#g(x)=4xrArrg'(x)=4#

#h(x)=(1-4x^2)^(1/2)rArrh'(x)=1/2(1-4x^2)^(-1/2)xx(-8x)#

#color(white)(xxxxxxxxxxxxxxxxxx)=-4x(1-4x^2)^(-1/2)#

#rArrd/dx((4x)/sqrt(1-4x^2))#

#=(4(1-4x^2)^(1/2)+16x^2(1-4x^2)^(-1/2))/(1-4x^2)#

#=(4(1-4x^2)^(-1/2)(1-4x^2+4x^2))/(1-4x^2)#

#=4/(1-4x^2)^(3/2)#

#rArrd/dx(tan^-1((4x)/sqrt(1-4x^2))#

#=(1-4x^2)/(1+12x^2)xx4/(1-4x^2)^(3/2)#

#=4/((1+12x^2)(1-4x^2)^(1/2)#