Use implicit differentiation to find $y^{\prime}$ and then evaluate $y^{\prime}$ at the point $(2,7)$.
\[
y-7 x^{3}+7=0
\]
\[
\begin{array}{l}
y^{\prime}=\square \\
\left.y^{\prime}\right|_{(2,7)}=\square \text { (Simplify your answer.) }
\end{array}
\]
Answer
So, the final answers are \(y^{\prime} = \boxed{-21x^{2}}\) and \(y^{\prime}|_{(2,7)} = \boxed{-84}\).
Step 1 ：Differentiate the given equation implicitly with respect to x. This means we will treat y as a function of x and use the chain rule when differentiating y. The derivative is \(y^{\prime} = -21x^{2}\).
Step 2 ：Substitute the point (2,7) into the derivative to find the slope of the tangent line at that point. The value of the derivative at the point (2,7) is \(y^{\prime}|_{(2,7)} = -84\).
Step 3 ：So, the final answers are \(y^{\prime} = \boxed{-21x^{2}}\) and \(y^{\prime}|_{(2,7)} = \boxed{-84}\).
