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.
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.
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
If f(x) = x cos x, find f ‘’(x).
Using the Product Rule, we get
To find f ‘’(x) we differentiate f ‘(x):
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
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