If x^yy^x = 1, then what is $\frac{dy}{dx}$ at (1, 1) equal to?

If y = (x^x)^x, then what is the value of $\frac{dy}{dx}$ at x = 1?

If y = (1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)(1 + x¹⁶) then what is $\frac{dy}{dx}$ at x = 0 equal to?

If y = cos x ⋅ cos 4x ⋅ cos 8x, then what is $\rm \frac{1}{y}\frac{dy}{dx}$ at $\rm x = \frac{\pi}{4}$ equal to?