Second derivative calculation

6 Find the exact coordinates of the point on the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) at which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).

1. Given that

$$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } } ,$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }\).

5 The equation of a curve is \(y = x ^ { 2 } \ln ( 4 x - 3 )\). Find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point on the curve for which \(x = 2\).

1 Given that \(y = 4 x ^ { 2 } \ln x\), find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = \mathrm { e } ^ { 2 }\).

7 It is given that \(y = \tan 4 x\).

1. By writing \(\tan 4 x\) as \(\frac { \sin 4 x } { \cos 4 x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = p \left( 1 + \tan ^ { 2 } 4 x \right)\), where \(p\) is a number to be determined.
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = q y \left( 1 + y ^ { 2 } \right)\), where \(q\) is a number to be determined.