differentiate y=a^x^2-cos (1/x)

Question

This question came from icu university, it was rated at 6 marks

Answer 1

Let a=e^b (b=ln(a)) then a^x^2=(e^b)^x^2=e^bx^2. Now let z=bx^2 and p=e^z, dp/dz=e^z, dz/dx=2bx.

dp/dx=dp/dz*dz/dx=e^z.2bx=2bxe^bx^2=2xln(a)a^x^2.

Differential of -cos(1/x)=sin(1/x)*(-1/x^2)=-sin(1/x)/x^2.

So dy/dx=2xln(a)a^x^2-sin(1/x)/x^2.

Answer 2

y = a^x^2 - cos(1/x)

dy/dx = a^x^2ln(a)(d/dx)(x^2) - ( -sin(1/x)(d/dx)(x^-1)

         = 2xa^x^2*lna - sin(1/x)/(x^2)