Is the Derivative of a Function the Slope?

We must be cautious in calling the derivative of a function the slope.  Although the two concepts are clearly related, there are nuances to each that separate them.

Derivative and Slope: What’s the difference?

Let us start with the definition of each.

A derivative of a function is a representation of the rate of change of one variable in relation to another at a given point on a function.  

The slope describes the steepness of a line as a relationship between the change in y-values for a change in the x-values.

Clearly, very similar ideas.  But let’s look at the important differences.  A function’s derivative is a function in and of itself.  It may be a constant (this will happen if our function is linear) but it may very well change between values of x.

Let f(x) = x².  Our derivative f’(x) = 2x.  If we take a look at the graph of x², we can see that for each step we take along the curve, the value of y changes more and more.  Between x = 0 and x = 1, y only increases by 1.  But between x = 1 and x = 2, y increases by 3.  If we keep going with this trend, between x = 2 and x = 3, y changes by 5.  We don’t have a constant change between equally spaced values of x, but rather y changes by twice as much each step.

A slope has the same idea, but can only be used for a line.  The slope of a line tells us how much that line’s y value changes for any given change in x, but we do not use this term for curves or non-linear functions as by definition, our slope is constant: A line always has the same slope.  Every step we take along the x-axis, the change in our value of y remains constant.  A positive slope indicates that y increases as x increases.   A negative slope implies that y decreases as x increases.  And a 0 slope implies that y is constant.  We cannot have the slope of a vertical line (as x would never change).

A function does not have a general slope, but rather the slope of a tangent line at any point.  In our above example, since the derivative (2x) is not constant, this tangent line increases the slope as we walk along the x-axis.  We cannot have a slope of y = x² at x = 2, but what we can have is the slope of the line tangent to this point, which has a slope of 4.

We can also take multiple derivatives, each gives us a new piece of information about our curve.  If the derivative of a function tells us how one variable changes with respect to another, the derivative of the derivative (named the second derivative or double derivative) tells us how about the change in the change of one variable with respect to another.  If we take the example above y = x², the derivative y’ = 2x shows us that the slope of a tangent line is constantly increasing.  The second derivative y’’ = 2 tells us that the change in this change is constant.

This is easier to see in a physical representation.  Let us give the position of a function as x(t) = 3t²-2t+1.  We can see that the position is not linear.  The derivative of this function x’(t) = 6t -2 gives us our velocity at any give time.  Our velocity we can see is also itself changing with time.  If we take our second derivative x’’(t) = 6 shows us how our velocity is changing with time.  This is named our acceleration, which in our above example is constant.

It is important to remember how to use the derivative to find the slope of a tangent line, but remember that the derivative itself is not a slope in and of itself.  The derivative is a powerful idea that use used in many different ways.