I am currently using the GGM05C Stokes' coefficients to reconstruct the gradient of gravity potential of Earth.
I have found an expression for said gradient in spherical coordinates in this technical report by the ICGEM. In particular, equation 122 on page 23 shows the partial derivatives of the gravity potential $W$ with respect to the three spherical coordinates parameters $r$ (distance to center), $\lambda$ (longitude) and $\varphi$ (geocentric latitude).
These equations involve the use of associated Legendre functions $P_{lm}$, which are a function of the latitude $\varphi$. I understand then that, when performing the partial derivatives with respect to $r$, $\varphi$ and $\lambda$, the Legendre functions remain unaltered in the derivatives with respect to $r$ and $\lambda$, since the Legendre functions are not a function of either $r$ or $\lambda$. However, since they are a function of the latitude $\varphi$, when we calculate $\dfrac{\partial W}{\partial \varphi}$, we need to derive the associated Legendre functions, obtaining the following expression, as indicated in equation 122 of the linked document:
$$ \dfrac{\partial W}{\partial \varphi} = \frac{GM}{r}\sum_{\mathscr{l}=0}^{\mathscr{l}_{max}}\left(\frac{R}{r}\right)^\mathscr{l}\sum_{m=0}^{\mathscr{l}}\dfrac{\partial P_{\mathscr{l}m}(sin\ \varphi)}{\partial \varphi}\left(C_{\mathscr{l}m}^Wcos(m\lambda)+S_{\mathscr{l}m}^Wsin(m\lambda)\right) $$
However, following the chain rule, shouldn't this derivative also include a multiplication by $cos\ \varphi$, since that is the derivative of the $sin\ \varphi$ nested within $P_{\mathscr{l}m}(sin\ \varphi)$?
