#dy/dx = d/dx(x^2(sin x)^4 + x(cos x)^-2)# *(differentiate both parts)*

#<=>dy/dx = d/dx(x^2(sin x)^4) + d/dx(x(cos x)^-2)# *(derivative of sum is sum of derivatives)*

#<=>dy/dx = (d/dx(x^2)*(sin x)^4 + x^2*d/dx((sin x)^4)) + d/dx(x(cos x)^-2)#*(product rule)*

#<=>dy/dx = (2x(sin x)^4 + x^2*4*(sin x)^3*d/dx(sin x)) + d/dx(x(cos x)^-2)# *(power rule and chain rule)*

#<=>dy/dx = (2x(sin x)^4 + 4x^2(sin x)^3*cos(x)) + d/dx(x(cos x)^-2)# *(trigonometric derivative)*

#<=>dy/dx = (2x(sin x)^4 + 4x^2(sin x)^3*cos(x)) + (d/dx(x)*(cos x)^-2 + x*d/dx((cos x)^-3))# *(product rule)*

#<=>dy/dx = (2x(sin x)^4 + 4x^2(sin x)^3*cos(x)) + ((cos x)^-2 + -3*x*(cos x)^-2*d/dx(cos x))# *(power rule and chain rule)*

#<=>dy/dx = (2x(sin x)^4 + 4x^2(sin x)^3*cos(x)) + ((cos x)^-2 + -3*x*(cos x)^-2*sin(x))# *(trigonometric derivative)*

So the derivative is:

#2x(sin x)^4 + 4x^2(sin x)^3*cos x + (cos x)^-2-3x(cos x)^-2*sin x#

You can try simplifying this, but I don't think there's much to simplify.