subject: Learning Second Derivative [print this page] In calculus (a branch of mathematics) the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity. The process of finding a derivative is called differentiation. Second derivative is higher derivatives of the function. (Source: Wikipedia)
Example Problems for Learning Second Derivative
Learning second derivative example problem 1:
Find the second derivative of the given function f (x) = 5x^3 + 24x^2 - 9.
Solution:
Given function is f (x) = 5x^3 + 24x^2 - 9
Differentiate the given function with respect to x, we get
f'(x) = 15x^2 + 48x
Again differentiate the above function with respect to x, we get
Find the second derivative of the given function f (x) = 12x^4 + 3x^2 + 7.
Solution:
Given function is f (x) = 12x^4 + 3x^2 + 7
Differentiate the given function with respect to x, we get
f'(x) = 48x^3 + 6x
Again differentiate the above function with respect to x, we get
f''(x) = 144x^2 + 6
Answer:
Second derivative value is f''(x) = 144x^2 + 6
Learning second derivative example problem 3:
Find the second derivative of the given function f (x) = 6x5 + 67x^3 + 34x^2
Solution:
Given function is f (x) = 6x5 + 67x^3 + 34x^2
Differentiate the given function with respect to x, we get
f'(x) = 30x^4 + 201x^2 + 68x
Again differentiate the above function with respect to x, we get
f''(x) = 120x^3 + 402x + 68
Answer:
Second derivative value is f''(x) = 120x^3 + 402x + 68
Practice Problems for Learning Second Derivative
Learning second derivative practice problem 1:
Find the second derivative of the given function f (x) = 26x^5 + 6x^3 + 9x
Answer:
Second derivative value is f''(x) = 520x^3 + 36x
Learning second derivative practice problem 2:
Find the second derivative of the given function f (x) = 2x^3 + 6x^2 + 3
Answer:
Second derivative value is f''(x) = 12x + 12
Learning second derivative practice problem 3:
Find the second derivative of the given function f (x) = 7x^3 + 13x
Answer:
Second derivative value is f''(x) = 42x
The second derivative of a function measures the concavity of the graph of . A function whose second derivative is positive will be concave up (sometimes referred to as convex), meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (sometimes called simply concave), and its tangent lines will lie above the graph of the function.
[edit]Inflection points
Main article: Inflection point
If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.
[edit]Second derivative test
Main article: Second derivative test
The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e. a point where ) is a local maximum or a local minimum. Specifically,
If then has a local maximum at .
If then has a local minimum at .
If , the second derivative test says nothing about the point , a possible inflection point.
The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration.
