Higher Derivatives and their Signs in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

The sign for the first derivative f ' of a differentiable function f tells us whether the graph of f is rising or falling. The sign for the second derivative of f, which is notated f '', tells us whether the curve y = f(x) is bending upward or downward.

The second derivative of f is the derivative of f ' ; it is denoted by

f '', and its value at x is:

And the third derivative f ''' of f is:

or notated also as:

When we begin with the function f and differentiate it n times in succession, we obtain the nth derivative

of f, with

If y = f (x), then the first n derivatives are written in operator notation as:

in function notation as:

and in differential notation as:

Below we see the history of the metamorphosis of the notation:

For example, we want to find the first four derivatives of

We will remove the fractional notation:

We then take a value from the exponents and multiply them to the coefficients for the first derivation:

and so on for the next three derivatives:

When the second derivative is positive, the curve bends upward, that is, the tangent line is turning counterclockwise [more formally, if f '' (x) > 0 on the interval I, then the first derivative is an increasing function on I, because its derivative f '' is positive]:

The curve can be considered bending upward even if it is not rising, so long as the tangent movement is counterclockwise:

However if f '' (x) <>I, then the first derivative f ' is decreasing on I, so the tangent line turns clockwise as x increases. In this case, curve y = f (x) is bending downward:

and also bending downward:

So:

For example, the graph below

shows the function above.

So, because

we can determine that

This is why the figure bends downward between negative infinity and 1, and it bends upward from 1 to infinity.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.243a-244b.