The derivative of the derivative of a given function is the second derivative. Given a function f(x), the 2nd derivative is represented as f’’(x) of [f(x)]. The definition of second derivative of a function f(x) at a point a is, f’’(x) = lim(x?a [f’(x) – f(a)]/(x-a)
It gives how fast the rate of change of a function is changing. It gives the information about the concavity.
Second Derivative Curvature
The curvature of a given function is described by the second-derivative of the function. If the function curves upwards then the curvature of the function is concave up and if the function curves downwards then the curvature of the function is concave down.
The function behavior corresponding to its 2nd derivative can be given as follows:
If f’’(x)>0, then the curvature of the function f(x) is concave up
If f’’(x)<0 br="" concave="" curvature="" down="" f="" function="" is="" of="" the="" then="" x="">If f’’(x)=0, then it corresponds to a possibility of an inflection point
The inflection point is the point where there is a change in the concavity of the function f(x).
Calculate Second Derivative
While calculating the 2nd derivative the 1st derivative becomes the function.
The two basic steps involved in finding the second of the derivative are as follows:
Step1: first the derivative of the function is found. The result thus obtained becomes the function to find it
Step2: the derivative of the first derivative gives the required derivative
If y=4x3- 3x^2 + 5x
Step1: The first derivative is given by y’ = [4x3 – 3x^2 + 5x]. Taking derivative of each of the terms,
y’= [4x3] – [3x^2] + [5x]
= 12x^2 – 6x + 5
Step2: Now the function is y’= 12x^2 – 6x + 5. The second-derivative would be,
y’’ = [y’]
= [12x^2 – 6x + 5] taking derivative of each of the terms,
= [12x^2] – [6x] + [5]
= 24x – 6 + 0
So, y’’ = 24x- 6 required 2nd derivative of y
Acceleration Second Derivative
Instantaneous velocity of a particle along a line at time t is the first derivative of a function which represents its position along a line at a particular time t. The derivative of this velocity is the second-derivative of the function which is the instantaneous acceleration of the particle at a time t.
For instance, let y= p(t) is the position of the particle, then the instantaneous velocity which is the first derivative is v=p’(t)and the instantaneous acceleration of the particle at time t which is the second derivative is a=v’(t) = p’’(t).
It gives how fast the rate of change of a function is changing. It gives the information about the concavity.
Second Derivative Curvature
The curvature of a given function is described by the second-derivative of the function. If the function curves upwards then the curvature of the function is concave up and if the function curves downwards then the curvature of the function is concave down.
The function behavior corresponding to its 2nd derivative can be given as follows:
If f’’(x)>0, then the curvature of the function f(x) is concave up
If f’’(x)<0 br="" concave="" curvature="" down="" f="" function="" is="" of="" the="" then="" x="">If f’’(x)=0, then it corresponds to a possibility of an inflection point
The inflection point is the point where there is a change in the concavity of the function f(x).
Calculate Second Derivative
While calculating the 2nd derivative the 1st derivative becomes the function.
The two basic steps involved in finding the second of the derivative are as follows:
Step1: first the derivative of the function is found. The result thus obtained becomes the function to find it
Step2: the derivative of the first derivative gives the required derivative
If y=4x3- 3x^2 + 5x
Step1: The first derivative is given by y’ = [4x3 – 3x^2 + 5x]. Taking derivative of each of the terms,
y’= [4x3] – [3x^2] + [5x]
= 12x^2 – 6x + 5
Step2: Now the function is y’= 12x^2 – 6x + 5. The second-derivative would be,
y’’ = [y’]
= [12x^2 – 6x + 5] taking derivative of each of the terms,
= [12x^2] – [6x] + [5]
= 24x – 6 + 0
So, y’’ = 24x- 6 required 2nd derivative of y
Acceleration Second Derivative
Instantaneous velocity of a particle along a line at time t is the first derivative of a function which represents its position along a line at a particular time t. The derivative of this velocity is the second-derivative of the function which is the instantaneous acceleration of the particle at a time t.
For instance, let y= p(t) is the position of the particle, then the instantaneous velocity which is the first derivative is v=p’(t)and the instantaneous acceleration of the particle at time t which is the second derivative is a=v’(t) = p’’(t).
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