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### Which of the following describes the derivative function f′(x) of a quadratic function f(x)?

Which of the following describes the derivative function f′(x) of a quadratic function f(x)?

Which of the following describes the derivative function f′(x) of a quadratic function f(x)?

### Polynomial functions and derivative (2): Quadratic functions ^{[1]}

A quadratic function is a polynomial function of degree 2.. To find the x-intercepts we have to solve a quadratic equation

We are interested in studying the derivative of simple functions with an intuitive and visual approach. To study the derivative of a quadartic function we are going to follow the same approach that we use in the case of a linear function.

We can say that this slope of the tangent of a function at a point is the slope of the function.. The slope of a function will, in general, depend on x

### CC The Derivative of a Function at a Point ^{[2]}

Nathan Wakefield, Christine Kelley, Marla Williams, Michelle Haver, Lawrence Seminario-Romero, Robert Huben, Aurora Marks, Stephanie Prahl, Based upon Active Calculus by Matthew Boelkins. How is the average rate of change of a function on a given interval defined, and what does this quantity measure?

What is the derivative of a function at a given point? What does this derivative value measure? How do we interpret the derivative value graphically?. How are limits used formally in the computation of derivatives?

It is a generalization of the notion of instantaneous velocity and measures how fast a particular function is changing at a given point. If the original function represents the position of a moving object, this instantaneous rate of change is precisely the velocity of the object

### A Gentle Introduction to Function Derivatives ^{[3]}

The concept of the derivative is the building block of many topics of calculus. It is important for understanding integrals, gradients, Hessians, and much more.

You will also discover why the derivative of a function is a function itself.. – How to compute the derivative of a function based upon the definition

This tutorial is divided into three parts; they are:. – The definition and notation used for derivatives of functions

### 3.2: The Derivative as a Function ^{[4]}

– Define the derivative function of a given function.. – Graph a derivative function from the graph of a given function.

– Describe three conditions for when a function does not have a derivative.. – Explain the meaning of a higher-order derivative.

If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function

### Quadratic Functions ^{[5]}

251 #1-8, 10, 11, 15, 16, 18, 19, 21, 23, 24, 30, 33, 37, 38, 75. A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.

Parabolas may open upward or downward and vary in “width” or “steepness”, but they all have the same basic “U” shape. The picture below shows three graphs, and they are all parabolas.

A parabola intersects its axis of symmetry at a point called the vertex of the parabola.. This means that if you are given any two points in the plane, then there is one and only one line that contains both points

### 3.2 The Derivative as a Function – Calculus Volume 1 ^{[6]}

– 3.2.1 Define the derivative function of a given function.. – 3.2.2 Graph a derivative function from the graph of a given function.

– 3.2.4 Describe three conditions for when a function does not have a derivative.. – 3.2.5 Explain the meaning of a higher-order derivative.

If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function

### 2.4 The Derivative Function ^{[7]}

We have seen how to create, or derive, a new function $f'(x)$ from a function $f(x)$, summarized in the paragraph containing equation 2.1.1. Now that we have the concept of limits, we can make this more precise.

We know that $f’$ carries important information about the original function $f$. In one example we saw that $f'(x)$ tells us how steep the graph of $f(x)$ is; in another we saw that $f'(x)$ tells us the velocity of an object if $f(x)$ tells us the position of the object at time $x$

Most functions encountered in practice are built up from a small collection of “primitive” functions in a few simple ways, for example, by adding or multiplying functions together to get new, more complicated functions. To make good use of the information provided by $f'(x)$ we need to be able to compute it for a variety of such functions.

### CC Using Derivatives to Describe Families of Functions ^{[8]}

Nathan Wakefield, Christine Kelley, Marla Williams, Michelle Haver, Lawrence Seminario-Romero, Robert Huben, Aurora Marks, Stephanie Prahl, Based upon Active Calculus by Matthew Boelkins. Section3.4Using Derivatives to Describe Families of Functions

How can we construct first and second derivative sign charts of functions that depend on one or more parameters while allowing those parameters to remain arbitrary constants?. Mathematicians are often interested in making general observations, say by describing patterns that hold in a large number of cases

In the next part of our studies, we use calculus to make general observations about families of functions that depend on one or more parameters. People who use applied mathematics, such as engineers and economists, often encounter the same types of functions where only small changes to certain constants occur

### Second derivative ^{[9]}

In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as “the rate of change of the rate of change”; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time

The last expression is the second derivative of position (x) with respect to time.. On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph

The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:. The second derivative of a function is usually denoted .[1][2] That is:

### Derivative Calculator: Wolfram ^{[10]}

Wolfram|Alpha is a great calculator for first, second and third derivatives; derivatives at a point; and partial derivatives. Learn what derivatives are and how Wolfram|Alpha calculates them.

Here are some examples illustrating how to ask for a derivative.. Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator

Given a function , there are many ways to denote the derivative of with respect to . When a derivative is taken times, the notation or is used

### Sources

- http://www.matematicasvisuales.com/english/html/analysis/derivative/quadratic.html#:~:text=The%20derivative%20function%20of%20a%20quadratic%20function%20is%20a%20linear%20function.&text=As%20Fermat%20already%20knew%2C%20at,x%2Daxis%20at%20this%20value.
- https://mathbooks.unl.edu/Calculus/sec-1-3-derivative-pt.html#:~:text=Derivative%20at%20a%20Point,the%20derivative%20at%20a%20point.
- https://machinelearningmastery.com/a-gentle-introduction-to-function-derivatives/#:~:text=In%20very%20simple%20words%2C%20the,pictorial%20illustration%20of%20the%20derivative.&text=In%20the%20figure%2C%20%CE%94x%20represents,in%20the%20value%20of%20x.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/03%3A_Derivatives/3.02%3A_The_Derivative_as_a_Function
- http://dl.uncw.edu/digilib/Mathematics/Algebra/mat111hb/PandR/quadratic/quadratic.html
- https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function
- https://www.whitman.edu/mathematics/calculus_online/section02.04.html
- https://mathbooks.unl.edu/Calculus/sec-3-2-families.html
- https://en.wikipedia.org/wiki/Second_derivative
- https://www.wolframalpha.com/calculators/derivative-calculator/