Second Derivative

Question 1

What is a second derivative? (Definition)

Answer

The second derivative (or second order derivative) is the application of the derivative on the (first) derivative of a function.

The second derivative therefore measures the variation of the first derivative of the function.

$$ f´´(x) = \lim_{h \to 0} \frac{f(x+h)-2f(x)+f(x-h)}{h^{2}} $$

Question 2

How to calculate a second derivative?

Answer

Calculate the derivative of the function(also called the first derivative), then the derivative of the derivative (called second derivative)

Example: $$ f(x) = x^2+\sin(x) \\ f´(x) = 2 x+\cos(x) \\ f´´(x) = 2 - \sin(x) $$

In physics the second derivative is usually used for acceleration calculations, in economics it allows us to analyze phenomena linked to growth rates.

Question 3

What is the list of common second derivatives?

Answer

The second derivatives to know are:

Name	Function	Second Derivative
constant/number	$$ k \in \mathbb{R} $$	$$ 0 $$
variable (only)	$$ x $$	$$ 0 $$
power n (exponent)	$$ x^n $$	$$ n(n-1) x^{n-2} $$
inverse	$$ \frac{1}{x} $$	$$ \frac{2}{x^3} $$
square root	$$ \sqrt{x} $$	$$ -\frac{1}{4x^{3/2}} $$
natural logarithm	$$ \ln \|x\| $$	$$ -\frac{1}{x^2} $$
exponential	$$ e^x $$	$$ e^x $$
exponent x	$$ a^x $$	$$ a^x (\ln(a))^2 $$
sine	$$ \sin(x) $$	$$ -\sin(x) $$
cosine	$$ \cos(x) $$	$$ -\cos(x) $$
tangent	$$ \tan(x) $$	$$ \frac{2\tan(x)}{\cos^2(x)} $$

Question 4

How to write a second derivative?

Answer

A second derivative can be written $ f´´(x) $ or $ f^{(2)}(x) $ or $ \ddot{f} $ (double dot) or $ \frac{d^2f}{dx^2} $.

On dCode use f ' ' which is the most used notation (and the fastest to write).

Question 5

How to use the second derivative for a monotony table?

Answer

The second derivative is used to know the variation of the slope of the curve representing the function. For a given interval:

— a positive second derivative means an increase of the slope (convex function)

— a negative second derivative signifies a decrease in thought (concave function)

— a zero second derivative means a straight / straight curve

For a given point:

— a second derivative canceling with a change of sign means a point of inflection, the curvature of the graphical representation changes and is reversed. It is a stationary point that can be a maximum of the function or a minimum of the function.

Question 6

Which functions do not have a second order derivative?

Answer

Any function that is non-continuous, and / or non-differentiable in at least one point, does not have a second derivative. See the tools definition domains of a function, and the differentiation/derivative domain of a function calculator.

Second Derivative