For the expression a*x + b, find the partial derivative with respect to a and the partial derivative with respect to b.

Step-by-step solution

1. 1

   Decide which symbols are frozen. A partial derivative differentiates with respect to one variable while treating every other symbol as a constant. Here the expression contains three symbols, $a$, $b$ and $x$, so which one is "the variable" changes from part to part — and $x$ is a constant in both parts of this question.

2. 2

   Differentiate with respect to $a$. Treating $x$ and $b$ as constants, the term $ax$ is a constant multiple of $a$ and $b$ is a pure constant:

   $\frac{\partial}{\partial a}(ax)=x,\qquad \frac{\partial}{\partial a}(b)=0,$

   $\frac{\partial f}{\partial a}=x.$

   The answer is $x$, not $1$ and not $ax$: differentiating $ax$ with respect to $a$ leaves the coefficient of $a$, which is $x$.

3. 3

   Differentiate with respect to $b$. Now $a$ and $x$ are frozen, so the entire product $ax$ is a constant:

   $\frac{\partial}{\partial b}(ax)=0,\qquad \frac{\partial}{\partial b}(b)=1,$

   $\frac{\partial f}{\partial b}=1.$

4. 4

   Contrast with the derivative in $x$. For completeness, $\frac{\partial f}{\partial x}=a$ — the same expression yields three different answers depending on which symbol is the variable. Comparing all three is the fastest way to internalise what "hold the others constant" actually does.

5. 5

   Interpret the result. In a straight-line model $y=ax+b$ fitted to data, these two partials are precisely the components of the gradient used by least squares: changing the slope $a$ by a small amount moves the prediction by $x$ times that amount, while changing the intercept $b$ moves every prediction by the same amount, $1$ for $1$. That is why data points far from the origin dominate the estimate of the slope.