Second Derivative for a Function

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Derivative of Composite Function/Second Derivative

Suppose is a one-one function. Then, we have the following formula:.

Jump to: navigation , search. This article is about a differentiation rule , i.

Simplifies Second Derivative of Rational Function

View other differentiation rules Statement Simple version at a specific point Suppose is a one-one function and is a point in the domain of such that is twice differentiable at and where denotes the derivative of. Then, we have the following formula for the second derivative of the inverse function : Simple version at a generic point Suppose is a one-one function. Then, we have the following formula: where the formula is applicable for all in the range of for which is twice differentiable at and the first derivative of at is nonzero.

Why do we do this?

Answers to Questions

The purpose of the specific point version is to emphasize that the point is fixed for the duration of the definition, i. However, the definition or fact applies not just for a single point but for all points satisfying certain criteria, and thus we can get further interesting perspectives on it by varying the point we are considering.

This is the purpose of the second, generic point version. Plug in a value that lies in each interval to the second derivative; if f '' x is positive, the function is concave upwards for that interval, and if f '' x is negative, the function is concave downwards for that interval.

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As a note, any point at which the function changes concavity is called a point of inflection. Some textbooks and individual classes define them slightly differently, though: Some call any point that changes the concavity a point of inflection, and some do not include points of discontinuity, only points at which the function actually exists.

Remember, a point of inflection is not necessarily all critical numbers, only ones which result in a change of concavity. It should also be noted, we don't necessarily have to factor and simplify a function all the way to the end; if you can see that part of a function won't equal zero, then it's perfectly fine to leave it as is, since factoring and simplifying won't change that.


In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative. Second Derivative. (Read about derivatives first if you don't already know what they are!) A derivative basically gives you the slope of a function at any point.

Just be careful, and make sure you absolutely know it can't equal zero first! Determine the regions in which the following function is concave upward or downward. Concavity is simply which way the graph is curving - up or down.

Calculus for the rest of us!

The chain rule states that when differentiating a function that contains another function inside of it, you should differentiate the outside function while keeping the inside function intact and then multiply that by the derivative of the inside function. The second derivative "Gamma" w. Moving the mouse over it shows the text. Learn more. How do you use the second derivative test to find the local maximum and minimum for

It can also be thought of as whether the function has an increasing or decreasing slope over a period.