Partial Derivative of a Linear Combination

The partial derivative of a linear combination with respect to any of its variables is simply the coeffecient of that variable.

Consider that you have the linear combination $$ y = 3x_1 + 4x_2 + 5x_3 $$

The derivative wrt $x_1$ is $$ \frac{dy}{dx_1} = 3 $$

Notice:

1. The derivative wrt to $x_1$ does not depend on any of the other variables.
2. The derivative is simply the coeffecient of the variable that you are taking the derivative with respect to.

Let’s walk through the derivative wrt $x_2$ a little more slowly. To take the derivative wrt to $x_2$, we must make all other variables constants. If we make $x_1$ and $x_3$ constant, both of those terms go to 0 (since the derivative of a constant is 0). Thus we are only left with the term $4x_2$, the derivative of which is $4$.

Thus we can conclude:

1. The derivative wrt to any single variable of a linear combination is just the coeffecient of that variable.
2. That means that the derivative wrt to any single variable of a linear combination does not depend on any of the other variables!

Another reason to love linear functions (aka linear combinations)! Because taking their partial derivatives are so simple and computationally cheap!