Pirates & Revolutionaries: 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: