We can also take multiple derivatives, each gives us a new piece of information about our curve.
We cannot have a slope of y = x 2 at x = 2, but what we can have is the slope of the line tangent to this point, which has a slope of 4. 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 the slope of a vertical line (as x would never change).Ī function does not have a general slope, but rather the slope of a tangent line at any point. And a 0 slope implies that y is constant. A negative slope implies that y decreases as x increases. A positive slope indicates that y increases as x increases. Every step we take along the x-axis, the change in our value of y remains constant. 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. We don’t have a constant change between equally spaced values of x, but rather y changes by twice as much each step.Ī slope has the same idea, but can only be used for a line. If we keep going with this trend, between x = 2 and x = 3, y changes by 5. But between x = 1 and x = 2, y increases by 3. Between x = 0 and x = 1, y only increases by 1. If we take a look at the graph of x 2, we can see that for each step we take along the curve, the value of y changes more and more. It may be a constant (this will happen if our function is linear) but it may very well change between values of x. A function’s derivative is a function in and of itself. But let’s look at the important differences. The slope describes the steepness of a line as a relationship between the change in y-values for a change in the x-values.Ĭlearly, very similar ideas. Let us start with the definition of each.Ī 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. Derivative and Slope: What’s the difference? Although the two concepts are clearly related, there are nuances to each that separate them.
We must be cautious in calling the derivative of a function the slope.
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