Q:

Check my derivative

This is used for a project and it is pretty late, so I just want to make sure I derived $C_3(x)$ correctly. Thank you!

A:

By Chain rule,
$$\dfrac{d\sqrt{a^2+x^2}}{dx}=\dfrac{d\sqrt{a^2+x^2}}{d(a^2+x^2)}\cdot\dfrac{d(a^2+x^2)}{dx}=\dfrac{2x}{2\sqrt{a^2+x^2}}$$
$$\dfrac{d(\sqrt{b^2+(c-x)^2})}{dx}=\dfrac{d(\sqrt{b^2+(c-x)^2})}{b^2+(c-x)^2}\cdot\dfrac{d(b^2+(c-x)^2)}{dx}=?$$

A:

Derivative with respect to what? Is the $c_{L}$ a function? You are not giving much of a context.
Assuming $c_{L}$ and $c_{R}$ are constants and $b$ and $c$ are independent of $x$, you are asking for derivative of $\sqrt{a^2+x^2}$ with respect to $x$ and for derivative of $\sqrt{b^2+c^2+x^2-2cx}$ with respect to $x$. In other words, you are asking for derivative of $f(x)^{\frac{1}{2}}$ with respect to $x$.
Then, using standard derivative rules, it is $\frac{1}{2}f(x)^{-\frac{1}{2}}f'(x)$. Hence none of your two derivatives are correct.

