91Ó°ÊÓ

Find the tangent line to the parametric curve \(x=\varphi_{1}(t), y=\varphi_{2}(t)\) at the point corresponding to the given value \(t_{0}\) of the parameter. $$ \varphi_{1}(t)=t /\left(1+t^{2}\right), \varphi_{2}(t)=t^{3} /\left(1+t^{2}\right) \quad t_{0}=2 / 3 $$

Calculate the linearization \(L(x)=f(c)+\) \(f^{\prime}(c), \cdot(x-c)\) for the given function \(f\) at the given value \(c\) $$ f(x)=e(x-1) / x, c=1 $$

A function \(f\) is given. Calculate \(f^{\prime}(x)\). $$ f(x)=1 /(1+\sqrt{x}) $$

In Exercises \(41-44,\) find a polynomial whose derivative the given polynomial. \(7 x^{6}-4 x+6\)

The population of a colony of bacteria after \(t\) hours is \(B(t)=5000+6 t^{3}\). At what rate is the population changing after 2 hours?

Calculate the derivative of each of the expressions in Exercises 39-44 by applying both the Product and Quotient Rules. $$ \sin ^{2}(x) / x $$

Find a polynomial whose derivative the given polynomial. \(x^{9}-2 x^{3}-1\)

Use the specified value of \(c\) and the given information about \(f\) and \(g\) to compute \((g \circ f)^{\prime}(c)\). \(g(9)=-2, g^{\prime}(-2)=7, g^{\prime}(9)=-3, f(-2)=9, f^{\prime}(-2)=4\) \(f^{\prime}(9)=5, c=-2\)

A function \(f\) is given. Calculate \(f^{\prime}(x)\). $$ f(x)=\sqrt{1+x^{2}} $$

Find the tangent line to the parametric curve \(x=\varphi_{1}(t), y=\varphi_{2}(t)\) at the point corresponding to the given value \(t_{0}\) of the parameter. $$ \varphi_{1}(t)=t e^{t}, \varphi_{2}(t)=t+e^{t} \quad t_{0}=0 $$