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 \(y = 1 - 2 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

\includegraphics{figure_8} Fig. 8 shows the curve \(y = f(x)\), where \(f(x) = \frac{1}{e^x + e^{-x} + 2}\).

1. Show algebraically that \(f(x)\) is an even function, and state how this property relates to the curve \(y = f(x)\). [3]
2. Find \(f'(x)\). [3]
3. Show that \(f(x) = \frac{e^x}{(e^x + 1)^2}\). [2]
4. Hence, using the substitution \(u = e^x + 1\), or otherwise, find the exact area enclosed by the curve \(y = f(x)\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\). [5]
5. Show that there is only one point of intersection of the curves \(y = f(x)\) and \(y = \frac{1}{4}e^x\), and find its coordinates. [5]

Given \(y = e^{kx}\), where \(k\) is a constant, find \(\frac{dy}{dx}\) Circle your answer. [1 mark] $$\frac{dy}{dx} = e^{kx} \quad \frac{dy}{dx} = ke^{kx} \quad \frac{dy}{dx} = kxe^{x-1} \quad \frac{dy}{dx} = \frac{e^{kx}}{k}$$