91Ó°ÊÓ

Find the point(s) where the slope of a tangent line to the given curve has the given value. In Exercises \(35-38,\) solve the given problems. $$y=8 x-x^{2}, m_{\mathrm{tan}}=6$$

In Exercises \(31-34,\) find the derivative of each function by using the definition. Then determine the values for which the function is differentiable. $$y=\frac{5 x}{x-4}$$

Evaluate the second derivative of the given function for the given value of \(x\). $$f(x)=x-\frac{2}{x^{3}}, x=-1$$

Find the derivative of each function by using the definition. Then determine the values for which the function is differentiable. $$y=\frac{5 x}{x-4}$$

Find the derivative of each function by using the definition. Then determine the values for which the function is differentiable. $$y=\frac{3}{x^{2}-1}$$

s represents the displacement, and t represents the time for objects moving with rectilinear motion, according to the given functions. Find the instantaneous velocity for the given times. $$s=2 t^{3}-4 t^{2} ; t=4$$

Find the indicated instantaneous rates of change. The electric current \(i\) at a point in an electric circuit is the instantaneous rate of change of the electric charge \(q\) that passes the point, with respect to the time \(t\). Find \(i\) in a circuit for which \(q=30-2 t\)

In Exercises \(31-34,\) find the derivative of each function by using the definition. Then determine the values for which the function is differentiable. $$y=\frac{3}{x^{2}-1}$$

Evaluate the second derivative of the given function for the given value of \(x\). $$y=3 x^{2 / 3}-\frac{2}{x}, x=-8$$

Find the point(s) where the slope of a tangent line to the given curve has the given value. In Exercises \(35-38,\) solve the given problems. $$y=12 x-\frac{1}{3} x^{3} ; m_{\mathrm{tan}}=-4$$