Differentiation - the chain rule

By Martin McBride, 2023-09-25
Tags: chain rule first principles derivative
Categories: differentiation calculus


In this article, we will look at using the chain rule to differentiate a composite function.

Composite functions

It is quite common in mathematics to work with composite functions. A composite function takes the form:

Composite function

Where f and g are any two functions of a single variable. We call f the outer function, and g the inner function. Combining 2 functions this way is called composing the functions. The result is called a composite function.

We will use this alternative notation for composite functions, as it is a little clearer when we use the chain rule:

Composite function

This means exactly the same as the previous notation. We compose f and g, then apply the resulting composite fucntion to the value x,

Examples of composite functions

Here is an example of a composite function, the cosine of x squared:

cos of x squared

This function is composed of these 2 standard functions:

cos of x squared

Here is a graph of this function:

cos of x squared

This function is similar to the cosine function, but because the value of x squared changes more rapidly as the magnitude of x gets larger, the cycles of the function get closer together as we get further away from 0.

Here is a second example, e to the power sine of x:

exp sin x

This function is composed of another 2 standard functions:

exp sin x

Here is the graph, with the function shown in red (and the sine function shown in grey):

exp sin x

The periodicity of this function is the same as the sine function. The value of the sine function is altered by the exponential function, for example:

  • When the sine function has its smallest value, -1, then e to the power sine of x is equal to 1/e (approximately 0.3679).
  • When the sine function has its largest value, 1, then e to the power sine of x is equal to e (approximately 2.7183).

The chain rule

How do we differentiate a composite function? Provided f and g themselves are differentiable functions, we can use the chain rule. This can be simply stated as:

Chain rule

What does this mean?

Well, the first term is f' composed with g. In other words, we find the derivative of f and pass g(x) in as an argument.

The second term is simply g', the derivative of g.

The chain rule tells us that the derivative of f composed with g is the product of the terms above.

We will give some examples below, together with an intuitive explanation of the rule.

Chain rule example 1

Let's see how this works with the first example from before:

cos of x squared

To apply the chain rule we must first differentiate f, and apply it to g. The derivative of cosine is minus sine so when we apply this to x squared we have:

Chain rule

We must then differentiate g. The derivative of x squared is 2x:

Chain rule

Multiplying the terms gives the derivative of the original composite function:

Chain rule

Here is a graph of the original function (left) and its derivative (right). The stationary points of the original function are marked with dots. These correspond to the zero points of derivative, as you would expect:

cos of x squared

This doesn't prove that the graph on the right is the derivative of the graph on the left, but it is consistent with it being true.

Chain rule example 2

Next, we will look at the second example:

exp sin x

In this case, the derivative of the exponential function f is itself. Applying this to g, the result is the same as the original expression:

Chain rule

Differentiating the sine term g gives cosine x:

Chain rule

Again we multiply the 2 terms to find the derivative of the original composite function:

Chain rule

Here is a similar graph of the function (left) and its derivative (right), and again the stationary points are consistent with the zero points of the derivative:

exp sin x

Intuitive explanation of the chain rule

To gain some kind of intuition as to why the chain rule works, let's consider a very simple composite function:

sin 2x

Now let's do a simple substitution, u = 2x:

sin u

We can differentiate this with respect to u, the result of course is just cosine:

Derivative of sin u

What is the derivative with respect to x? To answer that, here is a graph of sin u and sin x:

sin u derivative

Since u = 2x, the graph of sin u is compressed by a factor of 2 along the x axis - the value of u changes twice as quickly as x.

This in turn means that the slope of the curve sin u is twice the slope of the curve sin x. So we can find the derivative of the curve:

sin 2x derivative

The derivative is multiplied by 2 because u changes twice as quickly as x.

What about a slightly more complex function:

sin x^2

We can do a similar substitution, but this time u is equal to x squared:

sin x^2

If we differentiate with respect to u we get:

sin x^2

Here is the graph of sin u and sin x:

sin u derivative

This time the graph gets more and more compressed as x increases. This is because as x increases, the rate of change of u versus x gets faster and faster. So we can't just use a fixed multiplier of 2, we need to find how fast u is changing for any given value of x.

The rate of change of u wrt x, of course, is simply the derivative du/dx. Which in this example is:

sin u derivative

So instead of multiplying by 2 (as we did before), in this example we multiply by 2x:

sin u derivative

This is identical to the result we would get using the chain rule.

Chain rule and polynomials

Polynomials like this are an interesting example:

Polynomail

We could differentiate this by first multiplying out the brackets:

Polynomail

We can then differentiate it in the normal way, giving:

Polynomail

This is ok for a squared term, but if we had a higher power then multiplying out the brackets could be quite tedious. An alternative is to treat it as a composite function and apply the chain rule:

Polynomail

The two functions f and g are:

Polynomail

If we differentiate f and compose it with g we get:

Polynomail

If we differentiate g we get:

Polynomail

Combining these using the chain rule, in the same way as the previous examples, gives:

Polynomail

Of course, this gives exactly the same result as the direct differentiation.

See also



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