There are five basic cases to learn:

1. If our model is a simple regression model, Y_i = b₁ + b₂ X_i + e_i, then the slope of this function with respect to X_i is given simply by b₂. This slope is the same at all values of X_i where we might want to compute a slope. Straight lines have constant slope.

2. If our model is a multiple regression model, Y_i = b₁ + b₂ X_i + b₃Z_i + e_i, then its slope with respect to X_i is still constant and equal to b₂. Its slope with respect to Z_i is constant and equal to b₃. Taking the derivative of a sum of terms yields the sum of the derivatives of each term. Since X_i appears only in the second term in the sum, and nowhere else, only the derivative of this second term with respect to X_i is relevant to the derivative of the whole expression with respect to X_i.

3. If we have elected to use a quadratic model to capture curvature in the relationship between E[Y_i] and X_i, then our regression model (if X is the only explanatory variable) will be Y_i = b₁ + b₂ X_i + b₃ X_i²+ e_i. In general, the derivative of x² with respect to x is 2x. Thus, the derivative of this function for Y_i with respect to X_i will be b₂ + 2(b₃)X_i. Again, we take the derivative term-by-term. The derivative of b₂X_i is just b₂ (the coefficient on X_i). The derivative of b₃ X_i² is b₃ times the derivative of X_i² with respect to X_i, which is just 2X_i, so the complete derivative of the third term in the sum above is just 2(b₃)X_i.

Note that this derivative (slope) is not constant everywhere over the surface of the function. It changes according to the value of X_i at which it is being assessed.

- In particular, if b₃ is a positive number, the slope of the function in the X_i direction gets larger as X_i gets larger. This could mean, as you move from left to right along the X-axis, that the slope starts out negative and then becomes positive after a certain point. This would make the function a parabolic curve opening at the top (i.e opening "upwards").

- If b₃ is a negative number, the slope of the function in the X_i direction gets smaller as X_i gets larger. This could mean, as you move from left to right along the X-axis, that the slope starts out positive and then becomes negative after a certain point. This would make the function a parabolic curve opening at the bottom (i.e. opening "downwards").

In either of these cases, it is often useful to solve for the level of X_i at which the slope changes sign. In the first case, it would give us the location along the X-axis of the smallest fitted value of Y. In the second case, it would give us the location along the X-axis of the largest fitted value of Y. We find the crucial value of X by setting the formula for the derivative equal to zero and solving for the value of X which makes this equality true:

Note that if the coefficients b₂ (on X_i in the regression equation) and b₃ (on X_i² in the regression equation) are of opposite signs, this crucial value of X_i^* will be a positive value. If both coefficients have negative signs, or both have positive signs, then X_i^* will be a negative number. Sometimes when we fit a quadratic form, the only relevant part of the parabolic curve that is implied is one of the "shoulders" of the curve.
 

NOTE: We have developed a very helpful Java Applet, called the 2D Function Studio to help visualize quadratic functions and their derivatives, for different values of the intercept, slope and quadratic term coefficients. This Applet forms the basis of a lab session currently under development.

4. If we use interaction terms, our model might be of the form

In this case, we usually need to consider both the slope of the function (surface) in the X-direction and the slope of the function in the Z-direction. If it were not for the interaction term (the b₄ X_iZ_i term), this would be an ordinary linear-in-variables model with constant derivatives (slopes) b₂ and b₃, as before. The presence of the interaction term means that the slopes in each direction are NOT constant, but depend on the level of the OTHER explanatory variable in the model.

For example, the slope of this function with respect to X_i is b₂ + b₄Z_i. Depending on the sign of the estimated coefficient b₄, the slope of the function in the X-direction will get either larger or smaller as Z_i differs.

The slope of this function with respect to Z_i is b₃ + b₄X_i. Again, depending upon the sign of the estimated coefficient b₄ (on the interaction term), the slope of this function in the Z-direction will get either larger or smaller as X_i differs.

Intuitively, the use of an interaction term change our regression model from a flat, planar surface, to a "twisted" surface, with the degree of "twist" depending on the size of b₄.
 
 

5. When the relationships between variables are curves, but the slopes of these curves do not change sign, it is sometimes appropriate to consider models that involve logarithms of the variables. Fully logarithmic models (with all variables logged) are convenient for economists because they exhibit constant elasticities of Y with respect to (each) X. Logarithmic models can be of the following basic forms:

The geometric shapes of these relationships can be inferred by making use of the fact that the derivative of log(x) is (1/x) times the derivative of x.

Thus, for the first case above (the LOG-LIN model), the slope of the function in the X-direction can be determined as follows:

Thus, the slope in the X-direction depends on the current fitted value of Y at that point on the regression line. The sign of b₂ will determine whether the slope increases or decreases as Y gets larger.

For the second case above (the LIN-LOG model), the slope of the function in the X-direction can be determined as follows:

Thus, the slope in the X-direction depends on the current value of X at that point on the regression line. The sign of b₂ will determine whether the slope increases or decreases as X gets larger.

For the third case above (the LOG-LOG model), the slope of the function in the X-direction can be determined as follows:

Thus, the slope in the X-direction depends on the current values of both Y and X at that point on the regression line. The sign of b₂ will determine what happens to this slope as Y and X change with movement along the regression line. Note that b₂ in this case is conveniently equal to the elasticity of Y with respect to X: the percent change in Y for a percent change in X.