Differentiating with respect to a sum

Question

Suppose I have 2 arbitrary functions $f(\theta)$ and $g(\theta)$, and I need an expression for $\frac{\partial f}{\partial \theta +\partial g}$. How would I do so? Is the above expression equivalent to $\frac{\partial f}{\partial \theta}+\frac{\partial f}{\partial g}$ ?

Any response will be appreciated. Many thanks in advance.

Edit: For context of the problem, I was deriving the angle of rotation of a beam element of arclength $ r d\phi$, undergoing axial displacements of $du_z$ and azimuth displacements of $du_\theta$. Then, the tangent of overall rotation angle as in this image should be given by $\frac{1}{r}\frac{\partial u_z}{\partial \phi +\partial u_\theta}$. However, it seems that this form cannot be solved in as a differential equation easily, even if only a numerical approximation is desired. Hence, I was looking to simplify this expression in anyway possible so that I arrive at a form that can be solved as an differential equation.

Answer 1

I am assuming that you have three functions $\phi,v,w$ depending on some variable $\theta$. (I renamed $u_z$ respectively $u_\theta$ in $v$ respectively $w$.)

The change of angle between $\theta$ and $\theta+t$ can be described via

$$\tan\beta(\theta,t)=\frac{v(\theta+t)-v(\theta)}{r(\phi(\theta+t)-\phi(\theta))+r(w(t+\theta)-w(\theta))},$$ i.e.

$$\beta(\theta,t)=\arctan\frac{1}{r}\frac{(v(\theta+t)-v(\theta))/t}{((\phi+w)(\theta+t)-(\phi+w)(\theta))/t}.$$

This means that that the momentary change of the angle $\alpha$ at $\theta$ is described by

$$ \begin{aligned} \alpha'(\theta)=\beta(\theta,0) &=\arctan\frac{1}{r}\frac{f'(\theta)}{g'(\theta)} \end{aligned} $$

for the function $f=v$ and $g=\phi+w$. Then, it holds

$$\alpha(\theta)=\alpha(t)+\int_{t}^\theta\arctan\frac{1}{r}\frac{f'(x)}{g'(x)}dx$$

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Notice, that $\arctan$ is continuous, such that the limit for $t\rightarrow 0$ can be interchanged. More specifially

$$ \begin{aligned} \alpha'(\theta)=\lim_{t\rightarrow 0}\beta(\theta,t) &=\lim_{t\rightarrow 0}\arctan\frac{1}{r}\frac{(v(\theta+t)-v(\theta))/t}{((\phi+w)(\theta+t)-(\phi+w)(\theta))/t}\\ &=\arctan\frac{1}{r}\lim_{t\rightarrow 0}\frac{(v(\theta+t)-v(\theta))/t}{((\phi+w)(\theta+t)-(\phi+w)(\theta))/t}\\ &=\arctan\frac{1}{r}\frac{f'(\theta)}{g'(\theta)} \end{aligned} $$

Hereby, the function $g$ is defined as $g(\theta):=\phi(\theta)+w(\theta)$.