The curve C has equation y=rac{x}{9+x^2} - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 6

We need to find (\frac{dy}{dx}) for the function (y = (1 + e^x)^{\frac{3}{2}}).

1. Differentiate the function:

   Using the chain rule, we have:

   $\frac{dy}{dx} = \frac{3}{2}(1+e^x)^{\frac{1}{2}} \cdot e^x$

2. Substitute (x = \frac{1}{2} \text{ln} 3):

   First, evaluate (1 + e^{\frac{1}{2} \text{ln} 3}):

   (e^{\frac{1}{2} \text{ln} 3} = \sqrt{3}), hence:

   $1 + e^{\frac{1}{2} \text{ln} 3} = 1 + \sqrt{3}$

   Now substitute into the derivative:

   $\frac{dy}{dx} = \frac{3}{2}(1 + \sqrt{3})^{\frac{1}{2}} \cdot e^{\frac{1}{2} \text{ln} 3} \quad = \quad \frac{3}{2}(1 + \sqrt{3})^{\frac{1}{2}} \cdot \sqrt{3}$

3. Calculate the final value:

   As a final step:

   $\frac{dy}{dx} = \frac{3}{2}(1 + \sqrt{3})^{\frac{1}{2}} \cdot \sqrt{3} = 3 \cdot \frac{\sqrt{3}}{2}(1 + \sqrt{3})^{\frac{1}{2}}$

   Using approximations, this yields approximately 18.