91Ó°ÊÓ

Use the Generalized Power Rule to find the derivative of each function. $$ g(z)=z^{2}\left(2 z^{3}-z+5\right)^{4} $$

a. Find the equation of the tangent line to \(f(x)=x^{2}-4 x+6\) at \(x=1\) b. Graph the function and the tangent line on the window \([-1,5]\) by \([-2,10]\).

Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(s)=\frac{s^{3}+1}{s-1} $$

Fever The temperature of a patient \(t.\) hours after taking a fever reducing medicine is \(T(t)=98+8 / \sqrt{t}\) degrees Fahrenheit. Find \(T(2)\), \(T^{\prime}(2)\), and \(T^{\prime \prime}(2)\), and interpret these numbers.

For each piecewise linear function, find: a. \(\lim _{x \rightarrow 4^{-}} f(x)\) b. \(\lim _{x \rightarrow 4^{+}} f(x)\) c. \(\lim _{x \rightarrow 4} f(x)\) $$ f(x)=\left\\{\begin{array}{ll} 2-x & \text { if } x<4 \\ 2 x-10 & \text { if } x \geq 4 \end{array}\right. $$

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \text { } f(x)=\frac{1}{x^{2}} $$

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=\left(\frac{x+1}{x-1}\right)^{3} $$

Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(x)=\frac{x^{2}-2 x+3}{x+1} $$

\(41-44\). For each function, find: a. \(\lim _{x \rightarrow 0^{-}} f(x)\) b. \(\lim _{x \rightarrow 0^{+}} f(x)\) c. \(\lim _{x \rightarrow 0} f(x)\) $$ f(x)=|x| $$

a. Find the equation of the tangent line to \(f(x)=x^{3}-3 x^{2}+2 x-2\) at \(x=2\) b. Graph the function and the tangent line on the window \([-1,4]\) by \([-7,5]\).