We will use the chain rule where we take the derivative of the outside and multiply it by the derivative of the inside.
We have f(x)=sqrt(tan(2-x^3) It's better to rewrite it as an exponent (tan(2-x^3))^(1/2)
So, first lets take the derivative of the color(blue)("outside") using the power rule:
color(blue)(1/2)tan(2-x^3)^color(blue)(1/2-2/2)=> 1/2tan(2-x^3)^(-1/2)
This is just one part of the answer.
Now let's take the derivative of the color(green)("inside")
1/2(color(green)tan(2-x^3))^(-1/2)
d/dxcolor(green)tan(2-x^3) => sec^2(2-x^3)(-3x^2)
We ended up using the chain rule on the color(green)("inside") too.
The derivative of the tanx is sec^2x but we also have to apply the chain rule here too because sec^2x has a function in the inside too. Take the derivative of the color(red)("function") inside sec^2color(red)((2-x^3)) which is where we got the -3x^2 from.
Now we just multiply both the derivatives of the outside and inside.
1/(2sqrt(tan(2-x^3)))xxsec^2(2-x^3)(-3x^2)
d/dxsqrt(tan(2-x^3) = (-3x^2sec^2(2-x^3))/(2sqrt(tan(2-x^3))