6.24. Find the derivative $y_{x}^{\prime}(x)$ of the functions $x(t)=\operatorname{ctg} t^{2}$, $y(t)=\operatorname{arcctg} t^{4}$.

$$
y_{x}^{\prime}(x)=\frac{\left(\operatorname{arcctg} t^{4}\right)_{t}^{\prime}}{\left(\operatorname{ctg} t^{2}\right)_{t}^{\prime}}=\frac{-\frac{1}{1+t^{8}} \cdot 4 t^{3}}{-\frac{1}{\sin ^{2} t^{2}} \cdot 2 t}=\frac{2 t^{2} \cdot \sin ^{2} t^{2}}{1+t^{8}}
$$
Solution. 

 

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