Differentiation rules

Say we have an expression:

which we wish to differentiate. Then, we pick out of the tool box the power rule. It tells us how to differentiate the components, , with being some constant.

Showing this handy rule using the definition of the derivative turns out to be a bit tricky, so we shall start off with squares and cubes. Then, we'll finish with showing the rule using the definition for .

Say we have the function:

How do we find its derivative?

Have a look at this square shape:

Its side length is , so the area is . Increasing the side length by a tiny bit , we wish to know how much this affects the area. Notice that this is actually the derivative of the square.

Cutting the new parts from our square, we get these components:

So the square size increases by two times . Thus, calling the square area , the small change as we increase by is:

As we mentioned when introducing differentials, we are free to move the around like a variable. So we drag it over to the other side.

Tada! The derivative of the square. And, as the area of the square is a function of its side length, we have now effectively found the derivative of :

Now, some of you may have asked yourselves why we so carelessly threw out the little square in the upper right corner.

It turns out that as we decrease sufficiently, expressions with squared differentials as good as disappear completely. This may sound unconvincing, but the area actually goes to zero when we make infinitely small.

Adding a dimension, consider:

In the previous section, could be thought of as the area as a function of its side length. Analogously, is the volume of a cube as a function of its side length .

As we increase the side length a tiny bit, the change in volume is made up by these three blocks, each with the area :

The total change in area is thus:

The three rectangles on the edges of the cube have the area , so because they contain the term , their volume goes to zero as we make infinitely small.

Now, dragging the in the equation above to the left hand side, we end up with the change in volume, as a function of the changing side length :

The two examples and follow the same pattern as we take the derivative: the exponent is copied to in front of the variable, and then decreased by one.

Increasing the dimension of the cube representing the function, we end up being more confused than enlightened. However, the pattern continues as we want to take the derivative of and so on.

This can be formalized by the following theorem.

The formula is valid for all and such that the expression makes sense as a real number.

Note that if , the exponent which we move down is zero. Thus, the derivative of is zero.

Using the definition of the derivative, we show that the formula is valid for :

To show the rule for , one way is to proceed in similar manners, however it quickly gets laborious with the number of terms increasing as grows in .

Trigonometric functions describe quantities that behave periodically, following regular patterns.

With this in mind, we should expect the rate of change, or derivative, of such functions to display a similar periodic behavior.

Think about how the temperature outside usually changes over the four seasons.

As we move from the spring when it is getting warmer, toward the fall when the temperature coincidentally tend to fall, we go from a positive rate of change to a negative one.

Some time in the middle of the summer, when it's as hot as it gets, the rate of which the temperature changes will then be .

The same will happen at the time of minimum temperature in the winter, as the rate of change goes from being negative to positive.

If we let represent the temperature at the time , the corresponding derivative function, which describes the rate of temperature change, will be . This is one of a handful fundamental trigonometric rules of differentiation:

Note that must be expressed in radians, the rules do not hold for angles measured in degrees!

All of the above theorems can be proven using the formal definition of the derivative, but doing so could be a painful experience. Instead, we recommend that you at least learn the first two relatively simple rules by heart.

The slides to follow will teach you techniques that can be used to derive the last four rules from the derivatives of sine and cosine, and the relations between the trigonometric functions.

Given a function , how would you differentiate the function?

Differentiating it with the definition would be quite a chore. Just try expanding ...

Well, is the product of two functions, and , right?

Now think about what the derivative represents. The derivative describes a rate of change. How does the product change?

If we increase by some small quantity , it will cause a change in as well as in . The changes in the composite functions in turn depend on and . So we'd expect the terms and to crop up somewhere.

Since we're dealing with a product, it'd also be logical if there was some multiplication going on here.

Thus, the rule for the derivative of a product is:

It appears all the time, so you should be able to give the right answer.

Since:

then we get:

We need the chain rule to calculate the derivative of functions which has a function as its argument. The generic expression of such a function is:

Let's imagine we are asked to compute the derivative of this:

Here, we'd have as the outer function and as the inner function.

We can compute such derivatives by juggling a bit with the definition of the derivative. However, armed with the chain rule, you can take the short cut, straight to a neat little formula:

Using Leibniz's notation, the rule can also be written like this:

A side note: some literature use the notation to mean . These two have the same meaning.

To get a feeling for why the chain rule is correct, let's consider a math professor, she strolls slowly through the university corridor at night, as the windows shake from an actual train traveling at 10 times her speed. The sky is suddenly lit by a shooting star, moving at 2000 times the speed of the train.

How much faster is the star, compared to the math professor?

We know the relative speed of the star compared to the train. We also know how fast the train is compared to the professor. Using Leibniz's notation for derivatives, we can write:

This is the chain rule. It's just a way of breaking the derivative into smaller, more manageable parts.

In the example, the velocities are constant. We could have done the calculation without knowing about the chain rule. However, as the derivatives get more complicated, it's equally valid and all the more powerful.

Let's have a look at the example we mentioned at the beginning. We take the derivative of step by step, keeping in mind that we defined and . We get:

Frequently, you will encounter functions with an inner function inside the inner function, or where the inner function is a product. Sometimes, the composition is even more complicated.

When this happens, do not despair. To take the derivative of a composed function, we apply the rules in sequence.

However, some initial confusion is common, and taking the derivative is an art that requires practice. The exercises are a good place to start for mastering this subject.

Say I give you a function which looks something like this:

and I ask you to take the derivative of it. What do you do?

You could for example serve me the solution on a plate, using the quotient rule:

This rule can be derived from the product rule in the following way:

We note that:

Using this fact, we write:

And that is the quotient rule again.

A small remark: in taking the derivative of we can use the chain rule: . The inner function is and the outer function is .

The most fundamental example of the quotient rule is when taking the derivative of:

As mentioned above, we can use the chain rule to find the derivative. We can also use the quotient rule.

The numerator function would then be . The derivative of is zero, so:

This example has its own name: the reciprocal rule.

Let and . Taking the derivative of the quotient, we get:

The numerator can be simplified with trigonometric rules, as . Thus, we get:

But, as we have seen when talking about trigonometric functions, . So we have just shown:

This can also be written as , if we do not use trigonometric identities when simplifying.

Let's dissect the following polynomial:

It's a sum of the functions and . Also, could be thought of as a constant value, , multiplied by the function .

How should we differentiate the polynomial? Luckily we've got differentiation rules!

There's, for example, a sum rule:

It works a bit like multiplication:

Given a function , the change in the function is wholly determined by the changes in the composite functions.

Given a constant , we've also got the rule

Just as in regular multiplication, you can extract a term:

Finally, the derivative of a constant is .

This makes sense in light of the geometric interpretation of the derivative. If you'd draw a tangent line to a constant function like , the tangent wouldn't have any slope.

Let's start with the sum rule. Rearrange the terms in the definition of the derivative, and you'll get:

When , we end up with:

Ta-da. That's the reason the sum rule works.

To find our second rule, we'll pretty much do the same thing:

Again, letting , we get

The same line of reasoning gives us the third rule. If , we'll just get:

And when , well then the result is .

We can deploy these rules on our polynomial . They give us:

Since the sum rule and the constant factor rule hold, mathematicians say differentiation is a linear operation.