d/(dx)(tan^(-1).((4x)/sqrt(1-4x^2)))=?ddx(tan1.(4x14x2))=?

1 Answer
Dec 22, 2017

4/((1+12x^2)(1-4x^2)^(1/2))4(1+12x2)(14x2)12

Explanation:

•color(white)(x)d/dx(tan^-1(f(x)))=1/(1+(f(x))^2)xxf'(x)xddx(tan1(f(x)))=11+(f(x))2×f'(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)