The Difference Between First and Second Derivatives

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Now on the question at hand.  First, some quick references for you. The first derivative of the function f(x), is the slope of the tangent line to the function at the point x:

If the first derivative f’ is positive, then the function f is increasing (pointing upwards)

If the first derivative f’ is negative, then the function f is decreasing (pointing downwards)

As for the second derivate, the second derivative of a function is the derivative of the derivative of that function:

If the second derivative f” is positive , then the function f is concave up (looks like a U shape) .

If the second derivative f” is negative , then the function f is concave down (looks like an upside down U shape)

The second derivative is like the movie Inception.  The first derivative tells us whether or not the function is increasing or decreasing.  The second derivative shows us whether or not the first derivative is increasing or decreasing. So the second derivative plays directly off of the first.

If the second derivative turns out to be positive, then the first derivative will be increasing. The slope of the tangent line to the function is increasing as x increases.  If the second derivative turns out to be negative, then the first derivative is decreasing.  The slope of the tangent line to the function is then decreasing as x increases

Here is some more in-depth information, but this should be a good start.  If you would like to see it in action, take a look:

 

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