Differentiate exponential functions

Differentiate, with respect to \(x\),

1. \(\mathrm { e } ^ { 3 x } + \ln 2 x\),
2. \(\left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\).

1 It is given that \(\mathrm { f } ( x ) = 3 x - \frac { 5 } { x ^ { 3 } }\).
Find

1. \(\mathrm { f } ^ { \prime } ( x )\),
2. \(\mathrm { f } ^ { \prime \prime } ( x )\),
3. \(\int \mathrm { f } ( x ) \mathrm { d } x\).

7

1. Write down an expression for the gradient of the curve \(y = \mathrm { e } ^ { k x }\).
2. The line L is a tangent to the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) at the point where \(x = 2\). Show that L passes through the point \(( 0,0 )\).
3. Find the coordinates of the point of intersection of the curves \(y = 3 \mathrm { e } ^ { x }\) and \(y = 1 - 2 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

7. \(\mathrm { f } ( x ) = x + \frac { \mathrm { e } ^ { x } } { 5 } , \quad x \in \mathbb { R }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve \(C\), with equation \(y = \mathrm { f } ( x )\), crosses the \(y\)-axis at the point \(A\).
2. Find an equation for the tangent to \(C\) at \(A\).
3. Complete the table, giving the values of \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\) to 2 decimal places.

   \(x\)	0	0.5	1	1.5	2
   \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\)	0.45	0.91

2 Given \(y = \mathrm { e } ^ { k x }\), where \(k\) is a constant, find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { k x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = k \mathrm { e } ^ { k x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = k x \mathrm { e } ^ { k x - 1 } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { k x } } { k }$$