Question Detail

We use the product rule for differentiation, $(uv)' = u'v + uv'$, where $u = 3x^2$ and $v = \cos^{-1}x$.First, find the derivatives of $u$ and $v$:$u' = \frac{d}{dx}(3x^2) = 6x$.$v' = \frac{d}{dx}(\cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}$.Now, apply the product rule:$\frac{d}{dx}(3x^2 \cos^{-1}x) = (6x)(\cos^{-1}x) + (3x^2)(-\frac{1}{\sqrt{1-x^2}})$$= 6x \cos^{-1}x - \frac{3x^2}{\sqrt{1-x^2}}$.Thus, option A is the correct answer.