Derivative of a function


The derivative of a mathematical function is the rate or rate of change of a function at a certain point. That is, how fast a variation is occurring.

From a geometric perspective, the derivative of a function is the slope of the line tangent to the point where x is located.

In mathematical terms, the derivative of a function can be expressed as follows:

In the formula, x is the point at which the variable takes the value of x. Likewise, h is any number. This will then be equal to zero because, as we see in the image above, we must calculate the limit of the function when h approaches zero.

It should be remembered that, in general, the derivative is a mathematical function that is defined as the rate of change of one variable with respect to another. That is, by what percentage one variable increases or decreases when another has also increased or decreased.

We must specify that the limit of a function is defined as its tendency (to what value it approaches) when one of its parameters (in this case h) approaches a certain value.

Examples of the limit of a function

We can better understand the limit of a function with some examples. Let's look at the following case:

In this case, it was not necessary to find the limit when h approaches zero, since the result of dividing f (x + h) -f (x) by h results in a natural number, and not an algebraic expression where we can find ah, as is the following case:

Let's now look at another example:

Then, we divide by h:

Finally, I find the limit when h approaches 0:

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