LeSSOn223.8 Implicit differentiation: Probien (1 point) Find the derivative of \( y=\cos ^{-1}\left(8 x^{9}\right) \) (Use symbolic notation and fractions where needed.) \( y^{\prime}=\square \) help (fractions)

To find the derivative of \( y = \cos^{-1}(8x^9) \) using implicit differentiation, we will differentiate both sides with respect to \( x \). The derivative of \( \cos^{-1}(u) \) is given by \( -\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \). Setting \( u = 8x^9 \), we first find \( \frac{du}{dx} = 72x^8 \). Therefore, we have: \[ y' = -\frac{1}{\sqrt{1 - (8x^9)^2}} \cdot 72x^8 \] Simplifying this, we get: \[ y' = -\frac{72x^8}{\sqrt{1 - 64x^{18}}} \] In conclusion, the derivative is: \( y' = -\frac{72x^8}{\sqrt{1 - 64x^{18}}} \)