Use Newton's method to estimate the solutions of the equation \(x ^ { 2 } + x - 1 = 0\) . Start with \(x _ { 0 } = - 1\) for the left-hand solution and with \(x _ { 0 } = 1\) for the solution on the right. Then, in each case, find \(x _ { 2 }\) .

\(f ( x ) = \left \{ \left . \begin{array} { c c } { c } & { x = - 2 } \\ { 9 } & { x = 1 } \\ { \frac { x ^ { 2 } - 4 } { ( x + 2 ) ( x - 3 ) } } & { \text { otherwise } } \end{array} \right . \text { continuous at } x = - 2 ? \right .\)

(A) Differentiate the given equation implicitly and then solve for \(y ^ { \prime } \) . 

(B) Solve the given equation for \(y\) and then differentiate directly. 

\(5 x ^ { 3 } - 8 y - 6 = 0\) 

(A) First, differentiate \(5 x ^ { 3 } - 8 y - 6 = 0\) implicitly, term by term. 

\(\frac { d } { d x } ( 5 x ^ { 3 } ) = \square \) 

5. The table below gives the values of \(f ( x ) , f ^ { \prime } ( x ) , g ( x ) \) , and \(g ^ { \prime } ( x ) \) for various values of \(x\) . Use the table to answer the following questions. 

(a) Let \(A ( x ) = \ln ( g ( x ) ) \) . What is \(A ^ { \prime } ( 1 ) \) ? 

(b) Let \(B ( x ) = f ( x ) ( g ( x ) ) ^ { 2 } \) . What is \(B ^ { \prime } ( 2 ) \) ? 

(c) Let \(C ( x ) = \frac { ( f ( x ) ) ^ { 3 } } { g ( x ) } .\) What is \(C ^ { \prime } ( 3 ) \) ? 

5. The table below gives the values of \(f ( x ) , f ^ { \prime } ( x ) , g ( x ) \) , and \(g ^ { \prime } ( x ) \) for various values of \(x\) . Use the table to answer the following questions. 

(a) Let \(A ( x ) = \ln ( g ( x ) ) \) . What is \(A ^ { \prime } ( 1 ) \) ? 

(b) Let \(B ( x ) = f ( x ) ( g ( x ) ) ^ { 2 } \) . What is \(B ^ { \prime } ( 2 ) \) ? 

(c) Let \(C ( x ) = \frac { ( f ( x ) ) ^ { 3 } } { g ( x ) } .\) What is \(C ^ { \prime } ( 3 ) \) ? 

5. The table below gives the values of \( f ( x ) , f ^ { \prime } ( x ) , g ( x ) \) , and \( g ^ { \prime } ( x ) \) for various values of \( x \) . Use the table to answer the following questions. 

(a) Let \( A ( x ) = \ln ( g ( x ) ) \) . What is \( A ^ { \prime } ( 1 ) \) ? 

(b) Let \( B ( x ) = f ( x ) ( g ( x ) ) ^ { 2 } \) . What is \( B ^ { \prime } ( 2 ) \) ? 

(c) Let \( C ( x ) = \frac { ( f ( x ) ) ^ { 3 } } { g ( x ) } . \) What is \( C ^ { \prime } ( 3 ) \) ?