Techniques for Differentiation

Introduction

The previous Learn page on differentiation showed the basic principles of differentiation, and how it was derived from first principles. This page takes differentiation further, allowing you to differentiate many more complex functions, and introduces implicit differentiation.

Differentiating Trig, Exponential and Logarithmic Functions

The derivatives of the following functions should be learnt:
These can be proved, however the only important function to be able to differentiate yourself is the tan function, in which you will need the quotient rule.

The Chain Rule

The first of the rules we can use to allow us to differentiate more functions is called the chain rule. The chain rule allows you to differentiate composite functions. But what is a composite function?

A composite function is in essence a "function inside a function", this is applying a function to another function, for example:

fg(x) is a composite function and requires the chain rule in order to differentiate it. So how do you apply the chain rule?

Step 1: Call the function within the second function "u", and substitute "u" in to the original equation:

Step 2: Differentiate "u" with respect to x, and the function you were left with after substituting "u" in with respect to "u":
Step 3: Multiply the two derivatives you just worked out together:
Step 4: Substitute back for "u", and you've found the derivative! You've applied the chain rule succesfully:

The following is a general rule for applying the chain rule:

This Video is a quick 60-second summary of the chain rule, and is great for revising the rule, and for re-capping it:



The Product Rule

The product rule is used when the derivative of the product of two functions must be differentiated, for example:
So how do you apply the product rule?

Step 1: Call one of the functions "u" and the other "v":

Step 2: Differentiate both "u" and "v" with respect to x:
Step 3: Multiply "u" by the derivative of "v", and add the product of "v" and the derivative of "u":
In general the product rule is given by the following:

The Quotient Rule

The quotient rule is used to differentiate fractional functions. This means a function divided by another function. For example:
But how do you apply the quotient rule?

Step 1: Call the function in the numerator "u", and the function in the denominator "v":

Step 2: Differentiate "u" with respect to x, and differentiate "v" with respect to x:
Step 3: This step is somewhat similar to the product rule, take "v" and multiply it by the derivative of "u", then take away the product of "u" and the derivative of "v":
Step 4: Divide the result of the previous step by "v" squared, and that's it you've successfully applied the quotient rule:

The following is a general rule for applying the quotient rule:

**You may have noticed that the function that I gave as an example is equal to tanx. This example also, therefore, acts as a proof for the derivative of tanx I gave at the top of the page.

This Video is a quick 60-second summary of the quotient rule, and is great for revising the rule, and for re-capping it:



Implicit Differentiation

Sometimes you have functions in terms of x and y (or any two variables) and you cannot get it in the form y=f(x). To differentiate these functions you must differentiate implicitly. This requires the chain rule, and often requires the quotient rule and product rule, so make sure you've read the sections on these first. Let's take, for example the following function:
We need to differentiate both sides with respect to x but how do we differentiate functions with "y" in them with respect to x? It may not be obvious at first, but we will be using the chain rule for this:
So whenever we have a function in "y" and want to differentiate it with respect to x, we differentiate it with respect to "y" then multiply by dy/dx.

We can now differentiate both sides with respect to x (I use the product rule for xy):

All we have to do know is make dy/dx the subject of the formula, and we have differentiated the function, with respect to x:
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