Differentiation. The Chain Rule (Part I)

Differentiation. The Chain Rule

The Chain Rule is a method for finding the derivative of an output quantity of a function that is the composition of one or more functions. The Chain Rule allows us to find the derivatives of composite functions.

Here, we are going to learn an important theorem related to derivatives: the Chain Rule. After studying it, we will apply it to more complicated functions and then look at an inverse function.

Imagine we have two functions, f and g.

If we compose them (that is, perform the operations of g followed by f), we obtain another function: f circle g. The derivative of this composed function can be written as a single function (f circle g) with a dash here and is also equal to f(g(x)). It can also be written out as differentiating this composed function with respect to x.

The Chain Rule is a theorem that tells us that the derivative of the composition of two functions, f and g, is equal to the derivative of g evaluated at x multiplied by the derivative of f evaluated at x.

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Saying differently, the left-hand side is that one and the right-hand side is that one. Now pay attention.

It is important to note that we are not dealing with the product of two functions, f(x) and g(x), here. In other words, this is not the Product Rule. We are talking about the composition.

Let's consider the following example to illustrate the application of the chain rule.

So, use the Chain Rule to find the derivative of one over x squared plus seven cosine x.

First, we can consider this function as the composition of two functions. The first function is this one. The second function, f, is one over this, so according to the Chain Rule, to find the derivative of this is equal to the derivative of one over a function g; therefore, we differentiate it and get negative one — the function squared — evaluated at f.

Then we differentiate the function f. The derivative of this is simple: you get two x, so minus seven sine x. This is the final answer.

The derivative of an inverse function is defined as the inverse of the original function. If the graph of a function f(x) defined on an interval from a to b passes the horizontal line test, then f has an inverse g. The function f on this interval a, b has the inverse function g.

If you compose function f with function g, you get back the identity function; likewise if you compose g with f. The horizontal line test states that if you draw a horizontal straight line through the graph of y = alpha - where alpha lies between c and d - then every such line cuts the graph exactly once.

In this manner, we can define the function g on the interval a, b by reversing the process of defining f. The function g--the inverse of f--is denoted by f-1. We can see these two graphs here: blue is for y=f(x).

If you draw any horizontal line, y equals alpha, the inverse function can be determined. The horizontal line will always cut through the graph at exactly one point, thus defining the inverse function. If we take an alpha here, the value of the function at this point is given by this point.

If you graph y = f(x), moving the c, d interval onto the x-axis and the a, b interval onto the y-axis gives you the graph of its inverse function, g(x).

In the graph of the sine function, we see that the values are between -1 and 1. This graph passes the horizontal line test because every horizontal line between -1 and 1 will cut this graph at exactly one point. Therefore, we denote the inverse of sine by sin-1(x).

The range of values from minus pi/2 to pi/2 becomes the range of values from minus pi over 2 to pi over 2, which results in this graph.

Similarly, the cosine function between zero and pi has a value between one and minus one. It passes the horizontal line test and so the inverse function exists. The inverse of cosine is the same as putting this range minus one to one onto the x-axis and interval zero to pi onto the y-axis.

For the tangent function, it passes the horizontal line test and then its inverse function is given by this.

The inverse function for the cotangent x is defined as one over tangent x. Since when x equals to zero, tangent x is zero, so cotangent x becomes infinity; this is the point for x equals to zero. For this part, when x is between zero and pi over two, one over tangent x is here.

When x is greater than pi over two, the cotangent function equals this expression. In that range, y is between negative infinity and positive infinity and the graph passes the horizontal line test. Therefore, the inverse function for cotangent x is this one and its graph looks like this.