Why do we hold a variable constant when taking a partial derivative? What is the purpose of doing so? Is it acceptable to just implicitly differentiate, or does that cause problems? Also, how do you know/specify the direction of your tangent line/derivative?
Thanks!
For $f \,:\, \mathbb{R}^n \to \mathbb{R}^m$, you can view the partial derivative $\frac{\partial f}{\partial x_i}$ as a special case of a directional derivative $$ (D_cf)(x) := \lim_{\lambda \to 0} \frac{\|f(x + \lambda c) - f(x)\|}{\lambda} \text{.} $$
If $f$ is differentiable at $x$, i.e. if $f$ can be approximated linearly around $x$, then this linear approximation can be found from any set of directional derivatives $(D_{c_i}f)(x)$ at $x$ whose directions $c_i$ are linearly independent. The differential (or linear approximation) of $f$ at $x$ in the basis $(c_i)_{1 \leq i \leq n}$ is then $$ (dF)(x) = \begin{pmatrix} (D_{c_1}f)(x) &\ldots& (D_{c_n}f)(x) \end{pmatrix} \text{.} $$
If you use the canonical basis for the $c_i$, then $(D_{c_i}f)(x) = \frac{\partial f}{\partial x_i}$, and the differential of $f$ at $x$ in the canonical basis is thus $$ dF = \begin{pmatrix} \frac{\partial f}{\partial x_1} &\ldots& \frac{\partial f}{\partial x_n} \end{pmatrix} \text{.} $$
So for a differentiable $f$, the partial derivatives are simply a convenient choice for finding the differential $dF$. They are easy to compute because we can differentiate $x_1,\ldots,x_n$ individually, and they yield $dF$ in the canonical basis, which is probably what we want. But we could just as well use any other linearly independent set of directions.