91Ó°ÊÓ

Solve the given problems by finding the appropriate derivatives. Do the curves of \(y=x^{2}\) and \(y=1 / x^{2}\) cross at right angles? Explain.

Solve the given problems. Find \(d y / d x\) for \(y=\sqrt{x+1}\) by the method of Example \(5 .\) For what values of \(x\) is the function differentiable? Explain.

Find the derivative of \(y=1 / x^{3}\) as (a) a quotient and (b) a negative power of \(x\) and show that the results are the same.

Solve the given problems by finding the appropriate derivatives. If \(f(x)\) is a differentiable function, find an expression for the derivative of \(y=x^{2} f(x)\).

Find the acceleration of an object for which the displacement \(s\) (in \(\mathrm{m}\) ) is given as a function of the time \(t\) (in s) for the given value of \(t.\) $$s=\frac{16}{0.5 t^{2}+1}, t=2 \mathrm{s}$$

Solve the given problems by using implicit differentiation. The pressure \(P\), volume \(V\), and temperature \(T\) of a gas are related by \(P V=n(R T+a P-b P / T),\) where \(a, b, n,\) and \(R\) are constants. For constant \(V\), find \(d P / d T\).

Evaluate the indicated limits algebraically as in Examples \(10-14\). Change the form of the function where necessary. $$\lim _{x \rightarrow 1 / 3} \frac{9 x-3}{3 x^{2}+5 x-2}$$

Solve the given problems by finding the appropriate derivatives. If \(f(x)\) is a differentiable function, find an expression for the derivative of \(y=f(x) / x^{2}\).

Solve the given problems by using implicit differentiation. Oil moves through a pipeline such that the distance \(s\) it moves and the time \(t\) are related by \(s^{3}-t^{2}=7 t .\) Find the velocity of the oil for \(s=4.01 \mathrm{m}\) and \(t=5.25 \mathrm{s}\).

Find the acceleration of an object for which the displacement \(s\) (in \(\mathrm{m}\) ) is given as a function of the time \(t\) (in s) for the given value of \(t.\) $$s=250 \sqrt{6 t+7}, t=7.0 \mathrm{s}$$