Differentiate, with respect to x, (a) $e^{3x} + ext{ln} 2x,$ (b) $(5 + x^2)^{rac{1}{2}}.$ - Edexcel - A-Level Maths Pure - Question 4 - 2006 - Paper 4

For part (b), we will use the chain rule to differentiate the expression. Let:

$y = (5 + x^2)^{\frac{1}{2}}.$

Using the chain rule, we differentiate as follows:

1. The outer function is $u^{\frac{1}{2}}$, where $u = 5 + x^2$. The derivative of $u^{\frac{1}{2}}$ is $\frac{1}{2} u^{-\frac{1}{2}}$.
2. The derivative of $u$ with respect to $x$ is $\frac{d}{dx}(5 + x^2) = 2x$.

Putting it all together, we have:

$\frac{dy}{dx} = \frac{1}{2}(5 + x^2)^{-\frac{1}{2}} \cdot 2x = \frac{x}{(5 + x^2)^{\frac{1}{2}}}.$

Thus, the final answer for part (b) is:

$\frac{dy}{dx} = \frac{x}{(5 + x^2)^{\frac{1}{2}}}.$