Tool for Second Derivative calculation f''. The second derivative is the application of the derivation tool to the (first) derivative of a function, a double derivation on the same variable.
Second Derivative - dCode
Tag(s) : Functions
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The second derivative (or second order derivative) is the application of the derivative on the first derivative of a function. It therefore measures the variation of the first derivative of the function.
Calculate the derivative of the function(also called the first derivative), then the derivative of the derivative (called second derivative)
Example: $$ f(x) = x^2+\sin(x) \Rightarrow f´(x) = 2 x+\cos(x) \Rightarrow f´´(x) = 2 - \sin(x) $$
In physics the second derivative is usually used for acceleration calculations, in economics it allows us to analyze phenomena linked to growth rates.
The second derivatives to know are:
Name | Fonction | 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 | $$ \tan(x) $$ | $$ \frac{2\tan(x)}{\cos^2(x)} $$ |
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).
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.
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.
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Second Derivative on dCode.fr [online website], retrieved on 2024-07-27,