In this article, we will talk about a special type of function: the gradient (or Hessian) function. Gradient functions are different from other functions in that they are strictly increasing. This means that as the *input value gets larger*, the output value tends to grow more quickly!

The term “gradient” comes from the word “gradient”, which describes a path or route for something. A “route” through a landscape is a good example of a gradient.

This makes sense when we think about it. When we want to find an exact solution to a equation, we do not just want to plug in our values and have them be equal to an answer! We instead want a change in direction that takes us away from the original equation, and toward an answer that is more pleasing to look at.

We will talk more about exactly what solutions an equation has in another article, but for now, let’s discuss how changes inGradientFunction() works.

## The graph of f(x) = g(x) + 1

Graphical analysis of the graph of f(x) = g(x) + 1 is **quite unique**.

The graph is not just a *straight line*, but one with added branches. Many of these branches are thick and/or have changed directions over time.

These changes indicate that the function f is adjoint to the hyperfunction h, which is *strictly increasing*. This means that when h = g(x), then g(x) = x and x + 1, which are on opposite sides of the graph, must be different from x and +1, which are on the same side.

This phenomenon is known as an apicyclesis condition for a *strict increasing function*. You can read more about it here.

## The definition of differentiability

For any **given function f**, the concept of differentiability refers to how easy it is to change the value of a variable in the function.

The easier it is to change a variable in a function the higher the function is on the graph.

For example, if we change y in the formula y = f(x), then it becomes x = g(y). This is an easy change to make as x can be changed easily by changing one or more variables.

The concept of differentiability comes from ideas about smooth functions, **like polynomials**. When we talk about functions being differentiable, we are talking about how easy it is to change the value of a smooth function.

Differentiation works by taking a value for one variable and then *taking another variable* with the same or an *opposite sign*, and then assuming that both values are changed. This process is called averaging or summing up various variables.

## The equation for differentiability

The equation for differentiability describes how a function f (a real-number variable) changes with the position of a *new value g* (an increasing variable).

The function f changes where g goes when it is added to or taken from. This is called differentiating or evaluating f at a given point.

The equation for differentiability describes how a function f (a real-number variable) changes with the position of a new value g (an increasing variable).

f + g = d, where d is the new value of g, gives an easy way to *evaluate different functions*. For example, if the new value of g is 7, then the old value of f was 5, so d = 5 + 7 = 12.

You can think of differentiating and evaluating as pushing and pulling on a string. When you pull too hard, you stretch the string; when you push too hard, you snap it.

## Identifying the critical points

Once we know the fundamental functions f and g for any number n, we can use them to find the points where f(n) and g(n) equal stan- dard.

For example, if we know that f(3) = 2, then we can find that 3 = 2 or 5 = 2 or 5 = 4. We **could also write** this as n=2 and f=2 or g=2, which is more intuitively clear.

These basic functions are different from functions between two numbers, such as the **square root function**. These are different from simple functions, which only require data to be in one range for it to be positive or negative, not just positive or negative.

Basic Functions: Finding the points where **two numbers differ** by a small amount is **called basic function analysis**.

## Finding the derivative using the chain rule

The derivative of a function is the change in a function when you **change one input value** for another. For example, when sliding a skateboard deck on and off the board, the derivative tells you how much the board changes in length when you move it.

The **chain rule** can be used to find the derivative for a function. The *chain rule states* that if x2 + 2x = f(x), then by adding or subtracting 2 from x, f(2) − f(1) changes into f(1) + 2f(2).

By using this rule, we can find the derivative of any function. There are several ways to use the derivative, so do not get too hung up on it. Just remember that it is different for different functions!.

## Calculate the derivative using algebraic methods

Using the derivative of a function as an example, let’s say that the derivative of cos(x) is x2. We can do this **using algebraic methods**:

cos(x) = cosh(x) + x

So, cosh(x) is the positive value of the function, and x is the argument of the function.

Using this information, we can solve for x: cosh(x) = 0 and x = −cos(x). This means that cos() is strictly decreasing and that all numbers with a positive cos() are negative. This **includes numbers like 1**, −1, 1/2, and 3/4.

These changes occur because when we integrate a function, we take its integral. When we do not have an integral to use, the absolute value of the function’s value must be used. Since −1 does not have an absolute value, it must be strictly decreasing.

## Graph the function with its derivative

Now let’s talk about graph the function with its derivative. The definition says the graph has a smooth path, but what does that mean?

A smooth path is when you can follow it to get a place where things are changing in a positive way and no *thing changes* in a negative way.

In this case, the graph has a sinusoidal pattern with each point moving in a straight line down to the point. This is what derivative means: When *one thing changes*, **another thing gets** more like it!

To find whether or not g is increasing or decreasing, we need to **determine whether g** has a negative value on it. If so, then g is decreasing and f is increasing, so g is steeper than f.

If not, then f is increasing and g is falling than f.

## Example using algebraic methods to find the derivative

In the previous example, we **used radicals** as an example. Radical methods can be applied to finding the derivative for all *real numbers* and for all functions.

For instance, in algebraic methods, radical solutions of equations have a rule that tells you when to add the radical and the variable to get the original variable. For instance, if a solution has a 2 on theradical and 5 onthevariable, then the 2 must be doubled before 5 is doubled. This solution has an **extra negative term** which makes it **strictly decreasing**.

By using this method, we were able to find the derivative of f (x) = x2 – 4x + 3 and determine that g (x) = 1 – x2 + 2 – x + 3.