Second Derivative

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Second Derivative

If f' is the differential function of f, then its derivative f'' is also a function. The new function f'' is called the second derivative of f because it is the derivative of the derivative of f. Using the Leibniz notation, we write the second derivative of y = f(x) as

We can interpret f ‘’(x) as the slope of the curve y = f(‘(x) at the point (x, f ‘(x)). In other words, it is the rate of change of the slope of the original curve y = f(x). In general, we can interpret a second derivative as a rate of change of a rate of change. The most common example of this is acceleration.

Example:

The position of a particle is given by the equation
s = f(t) = t³ – 4t² + 5t
where t is measured in seconds and s in meters.

a) Find the velocity function of the particle
b) Find the acceleration function of the particle.

Solution:

a) The velocity function is the derivative of the position function.

b) The acceleration function is the derivative of the velocity function

The following diagrams show how the second derivative test van be used to find the local maximum and local minimum.

Second Derivative Test
How to use the second derivative test to identify the presence of a relative maximum or a relative minimum at a critical point? If the second derivative is positive at a critical number - a local minimum is present. If the second derivative is negative at a critical number - a local maximum is present.
To identify the relative extrema:
1. Find the first derivative, set it equal to zero and identify the critical numbers.
2. Plug the critical numbers into the second derivative function to determine the concavity of the function to see if its concave up or concave down. If it's concave up - it's a relative maximum. If it's concave down, it's a relative minimum. You can confirm the results of the second derivative test using the first derivative test with a sign chart on a number line.

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Example:

If f(x) = x cos x, find f ‘’(x).

Solution:

Using the Product Rule, we get

 

To find f ‘’(x) we differentiate f ‘(x):

Higher Derivatives

The third derivative f ‘’’ is the derivative of the second derivative. The third derivative can be interpreted as the slope of the curve or the rate of change of the second derivative.

The process can be continued. The fourth derivative is usually denoted by f⁽⁴⁾. In general the nth derivative of f is denoted by f⁽ⁿ⁾ and is obtained from f by differentiating n times.

Concavity and Second Derivatives
Examples of using the second derivative to determine where a function is concave up or concave down.

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Using Implicit Differentiation to find a Second Derivative

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