91Ó°ÊÓ

Find the derivative of each of the given functions. $$f(R)=\sqrt{\frac{2 R+1}{4 R+1}}$$

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

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

Evaluate the second derivative of the given function for the given value of \(x\). $$f(x)=\sqrt{x^{2}+9}, x=4$$

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{x \rightarrow 3}(3 x-2)$$

Find the indicated instantaneous rates of change. The distance \(s\) (in \(\mathrm{m}\) ) above the ground for a projectile fired vertically upward with a velocity of \(44 \mathrm{m} / \mathrm{s}\) as a function of time \(t\) (in s) is given by \(s=44 t-4.9 t^{2} .\) Find \(t\) for \(v=0\)

Evaluate the indicated limits by direct evaluation as in Examples \(10-14 .\) Change the form of the function where necessary. $$\lim _{x \rightarrow 4} \sqrt{x^{2}-7}$$

Evaluate the derivatives of the given functions at the given points. $$2(x+y)^{3}-y^{2} / x=15 ; \quad(4,-2)$$

Find the derivative of each of the given functions. $$y=\left(\frac{2 x+1}{3 x-2}\right)^{2}$$

Determine an expression for the instantaneous velocity of objects moving with rectilinear motion according to the functions given, if s represents displacement in terms of time \(t\). $$s=s_{0}+v_{0} t+\frac{1}{2} a t^{2}$$