91Ó°ÊÓ

In Exercises 65-70, calculate the derivative of the given expression. $$ x^{-6} $$

Graphically locate the points on the curve \(x^{3}-6 x y^{2}=4\) where the tangent line is vertical. Confirm that the method of implicit differentiation yields the same points.

Use a central difference quotient to approximate \(f^{\prime}(c)\) for the given \(f\) and \(c .\) Plot the function and the tangent line at \((c, f(c))\). $$ f(x)=\arcsin \left(\frac{x^{2}}{x^{2}+1}\right), \quad c=1.3 $$

Graph the given function \(f\) in the suggested viewing rectangle \(R\). From this graph, you will be able to detect at least one point at which \(f\) may not be differentiable. By zooming in, if necessary, identify each point \(c\) for which \(f^{\prime}(c)\) does not exist. Sketch or print your final graph, and explain what feature of the graph indicates that \(f\) is not differentiable at \(c\). $$ f(x)=x \sin ^{1 / 3}(x), R=[1,5] \times[-5,2.5 $$