1. The problem is to find the derivative of the function $$f(x)=\frac{1-\cos(0x)}{1-(1+\tanh^5(x))}$$ with respect to $x$. 2. Simplify the function first: - Since $\cos(0x) = \cos(0) = 1$, the numerator becomes $1-1=0$. - The denominator is $1-(1+\tanh^5(x)) = 1-1-\tanh^5(x) = -\tanh^5(x)$. 3. So the function simplifies to $$f(x) = \frac{0}{-\tanh^5(x)}=0$$ for all $x$ where the denominator is defined. 4. The derivative of a constant zero function is zero: $$f'(x) = 0$$ 5. Therefore, the derivative of the given function is zero everywhere it is defined. Final answer: $$f'(x) = 0$$