# How to Use a Partial Derivative to Measure a Slope in Three Dimensions

You can use a partial derivative to measure a rate of change in a coordinate direction in three dimensions. To do this, you visualize a function of two variables *z* = *f*(*x**,* *y*) as a surface floating over the *xy*-plane of a 3-D Cartesian graph. The following figure contains a sample function.

Now take a look at the function *z* = *y**,* shown here.

As you can see, this function looks a lot like the sloped roof of a house. Imagine yourself standing on this surface. When you walk parallel with the *y*-axis, your altitude either rises or falls. In other words, as the value of *y* changes, so does the value of *z**.* But when you walk parallel with the *x-*axis, your altitude remains the same; changing the value of *x* has no effect on *z**.*

So intuitively, you expect that the partial derivative

is 1. You also expect that the partial derivative

is 0.

Calculating partial derivatives isn’t much more difficult than evaluating regular derivatives. Given a function *z*(*x**,* *y*), the two partial derivatives are

Here’s how you calculate them:

To calculate

treat

*y*as a constant and use*x*as your differentiation variable.To calculate

treat

*x*as a constant and use*y*as your differentiation variable.

For example, suppose you’re given the equation *z* = 5*x*^{2}*y*^{3}. To find

treat *y* as if it were a constant — that is, treat the entire factor 5*y*^{3 }as if it’s one big constant — and differentiate* x*^{2}:

To find

treat *x* as if it were a constant — that is, treat 5*x*^{2} as if it’s the constant — and differentiate *y*^{3}:

As another example, suppose that you’re given the equation *z* = 2*e*^{x}* *sin *y* + ln *x**.* To find

treat *y* as if it were a constant and differentiate by the variable *x**:*

To find

treat *x* as if it were a constant and differentiate by the variable *y**:*

As you can see, when differentiating by *y**,* the ln *x* term is treated as a constant and drops away completely.

Returning to the earlier example — the “sloped-roof” function *z* = *y* — here are both partial derivatives of this function:

As you can see, this calculation produces the predicted results.