Answer
$f^{\prime}(1)=0$
Work Step by Step
$f(x)=\cos \left(\ln (x^{2})\right)\\
f^{\prime}(x)=-\sin \left(\ln (x^{2})\right) \times \frac{1}{x^{2}} \times 2 x\\
f^{\prime}(x)=-\frac{2 \sin \left(\ln (x^{2})\right)}{x}\\
f^{\prime}(1)=-\frac{2 \sin \left(\ln (1^{2})\right)}{1}\\
f^{\prime}(1)=0$
