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.

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

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

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.

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 | $$ \tan(x) $$ | $$ \frac{2\tan(x)}{\cos^2(x)} $$ |

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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Cite as source (bibliography):

*Second Derivative* on dCode.fr [online website], retrieved on 2024-10-07,

derivative,second,function,differentiation,calculator,acceleration,slope

https://www.dcode.fr/second-derivative

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