8 julio, 2024

Successive derivatives: definition, examples and solved exercises

What are successive derivatives?

The successive derivatives are the derivatives of a function after the second derivative. The process to calculate the successive derivatives is as follows: there is a function f, which we can derive and thus obtain the derivative function f’. We can derive this derivative of f again, obtaining (f’)’.

This new function is called the second derivative; all derivatives computed from the second are successive; These, also called higher order, have great applications, such as giving information about the plot of the graph of a function, the test of the second derivative for relative extremes and the determination of infinite series.

Definition

Using Leibniz’s notation, we have that the derivative of a function “y” with respect to “x” is dy/dx. To express the second derivative of «y» using the Leibniz notation, we write as follows:

In general, we can express successive derivatives as follows with Leibniz notation, where n represents the order of the derivative.

Other notations used are the following:

Some examples where we can see the different notations are:

Example 1

Obtain all the derivatives of the function f defined by:

Using the usual derivation techniques, we have that the derivative of f is:

By repeating the process we can obtain the second derivative, the third derivative, and so on.

Note that the fourth derivative is zero and the derivative of zero is zero, so we have:

Example 2

Calculate the fourth derivative of the following function:

Deriving the given function we have as a result:

speed and acceleration

One of the motivations that led to the discovery of the derivative was the search for the definition of instantaneous velocity. The formal definition is as follows:

Let y = f

Once the velocity of a particle is obtained, we can calculate instantaneous acceleration, which is defined as follows:

The instantaneous acceleration of a particle whose trajectory is given by y = f

Example 1

A particle moves along a line according to the position function:

Where «y» is measured in meters and «t» in seconds.

At what instant is its velocity 0?
At what instant is its acceleration 0?

By deriving the position function «y» we have that its velocity and acceleration are given respectively by:

In order to answer the first question, it is enough to determine when the function v becomes zero; this is:

We proceed with the following question in an analogous way:

Example 2

A particle moves along a line according to the following equation of motion:

Determine “t, y” and “v” when a = 0.

Knowing that the velocity and acceleration are given by

We proceed to derive and obtain:

Setting a = 0, we have:

From where we can deduce that the value of t for a to be equal to zero is t = 1.

Then, evaluating the position function and the velocity function at t = 1, we have that:

Applications

implicit derivation

Successive derivatives can also be obtained by implicit differentiation.

Example

Given the following ellipse, find «y»:

Differentiating implicitly with respect to x, we have:

Then, differentiating implicitly with respect to x again, gives us:

Finally, we have:

relative extremes

Another use that we can give to second order derivatives is in the calculation of relative extrema of a function.

The first derivative test for local extrema tells us that if we have a function f that is continuous on an interval (a, b) and there exists a c that belongs to that interval such that f’s vanishes in c (that is, that c is a critical point), one of these three cases can occur:

If f´(x) > 0 for any x belonging to (a,c) and f´(x)<0 for x belonging to (c,b), then f(c) is a local maximum.
If f´(x) < 0 for any x belonging to (a,c) and f´(x)>0 for x belonging to (c,b), then f(c) is a local minimum.
If f´(x) has the same sign in (a,c) and in (c,b), it implies that f(c) is not a local extremum.

Using the second derivative test we can know if a critical number of a function is a local maximum or minimum, without having to see what the sign of the function is in the aforementioned intervals.

The second drift test tells us that if f´(c) = 0 and that f´´(x) is continuous on (a, b), it happens that if f´´(c) > 0 then f(c) is a local minimum and if f´´(c) < 0 then f(c) is a local maximum.

If f´´(c) = 0, we cannot conclude anything.

Example

Given the function f(x)= x4 + (4/3)x3 – 4×2, find the relative maxima and minima of f by applying the second derivative test.

First we calculate f´(x) and f´´(x) and we have:

f´(x) = 4×3 + 4×2 – 8x

f´´(x) = 12×2 + 8x – 8

Now, f´(x) = 0 if and only if 4x(x + 2)(x – 1)= 0, and this happens when x=0, x=1 or x=– 2.

To determine if the critical numbers obtained are relative extremes, it is enough to evaluate in f´´ and thus observe its sign.

f´´(0) = – 8, so f(0) is a local maximum.

f´´(1) = 12, so f(1) is a local minimum.

f´´(– 2) = 24, so f(– 2) is a local minimum.

Taylor series

Let f be a function defined as follows:

This function has a radius of convergence R > 0 and has derivatives of all orders in (-R, R). Successive derivatives of f give us:

Taking x = 0, we can obtain the values ​​of cn as a function of its derivatives as follows:

If we take an = 0 as the function f (that is, f^0=f), then we can rewrite the function as follows:

Now consider the function as a power series at x = a:

If we carry out an analysis analogous to the previous one, we would have to write the function f as:

These series are known as Taylor series of f at a. When a = 0 we have the particular case called Maclaurin series. This type of series is of great mathematical importance, especially in numerical analysis, since thanks to these we can define functions in computers such as ex , sin(x) and cos(x).

Example

Obtain the Maclaurin series for ex.

Note that if f(x)= ex, then f(n)(x)= ex and f(n)(0) = 1, so its Maclaurin series is:

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