In this explainer, we will learn how to use the power rule of derivatives and the derivative of a sum of functions to find the derivatives of polynomials and general power functions.

We begin by recalling the definition of the derivative.

### Derivative

The derivative of a function is defined as at the points where the limit exists.

Using such a definition to calculate derivatives can be rather tedious. We would therefore like to find some rules which we can use to simplify finding the derivatives of functions. In this explainer, we will consider some of the key rules which will enable us to differentiate a whole collection of functions.

We will begin by considering one of the simplest categories of functions: constant functions.

### Example 1: Derivative of a Constant

Find , given that .

### Answer

Recall the definition of the derivative for a general function :

Substituting in , we have

Therefore, the derivative of is zero for all values of .

The last example demonstrated a general rule of derivatives that the derivative of a constant is zero.

### Derivative of a Constant Function

For a constant ,

We can now consider functions of the form .

### Example 2: The Power Rule for Positive Integers

- Find the derivative of by first principles.
- Find the derivative of by first principles.
- Find the derivative of by first principles.
- By considering the pattern, what is the derivative of ?

### Answer

**Part 1**

Recall the definition of the derivative for a general function :

Using the function , we have

Since , we can cancel the common factor of from the numerator and denominator to get

**Part 2**

Using the function and the definition of the derivative, we have

Expanding the parentheses in the numerator, we have

Since , we can cancel the common factor of from the numerator and denominator to get

**Part 3**

Using the function and the definition of the derivative, we have

Expanding the parentheses in the numerator, we have

Since , we can cancel the common factor of from the numerator and denominator to get

**Part 4**

Putting together the answers from the first three parts of the question, we have

We can see that when we take the derivative of a power of , the power reduces by one. In addition to this, we find that there is a constant equal to the original power. Hence,

The last question leads us to a general rule for differentiating powers of ; we call it the power rule.

### Power Rule (Positive Integers)

For any positive integer ,

To prove the power rule for positive integers, we will need to use the binomial theorem. Recall that the binomial theorem enables us to expand binomials raised to any positive integer power in particular:

Let ; using the definition of the derivative, we have

Using the binomial theorem, we can expand as follows:

Since , we can cancel the common factor of from the numerator and denominator to get

Taking the limit as , the only term without a positive power of in it is . Hence,

Notice that if we set , we have a constant function and the power rule tells us that the derivative is zero in agreement with our initial rule regarding the derivatives of constant functions.

Although this is a powerful rule that enables us to consider derivatives of a much larger class of functions, it is still rather limiting since we are not able to use this to differentiate functions made of multiple terms such as polynomials. We would, therefore, like to know rules about how to differentiate functions multiplied by constants and functions formed as the sum of two simpler functions. We will begin with considering the derivative of a function , where is a constant and is a differentiable function.

Using the definition of the derivative, we have

Factoring out the common factor of , we have

Given that is differentiable, we can use the rules of finite limits to rewrite this as

### Constant Multiple Rule

For a constant ,

We will now consider the derivative of a function defined as the sum of two differentiable functions. Let ; then, using the definition of the derivative, we have

Since both and are differentiable, we have

Using the rule in conjunction with the constant multiple rule, we can see

Using the sum rule, we have

We can now use the constant multiple rule to get

We summarize these results below.

### Sum and Difference Rule

For two differentiable functions and ,

We now know the necessary tools to be able to differentiate polynomials.

### Example 3: Derivatives of Polynomials

Given that , determine if .

### Answer

Since we know the value of the derivative at a certain point, we should begin by finding an expression for the derivative. Using the sum and difference rule, we have

We can now use the constant multiple rule to get

We can now apply the power rule, to each term as follows:

Since , we have

Therefore, .

The last example demonstrated how to carefully apply the sum, difference, and constant multiple rules. In general, we will not need to give the full details of applying these rules when differentiating functions. In later examples, we will not use the same level of detail when applying these rules.

In the next example, we will consider whether the power rule generalizes to include negative powers.

### Example 4: Differentiating Negative Powers

Find the derivative of .

### Answer

Using the definition of the derivative with , we have

We can express this as a single fraction as follows:

Since , we can cancel it from the numerator and denominator as follows:

Using the laws of finite limits, we have

Taking the limit as , we have

From the previous example, we saw that

This can also be written as which is the same form as the power rule if we set . This is not an accident; and in fact, the power rule generalizes to all integers.

### Power Rule (Integers)

For any integer ,

We will not prove this version of the power rule right now. However, it is straightforward to prove once we have been introduced to the quotient rule. In the next example, we will consider whether the power rule generalizes further to include fractional exponents.

### Example 5: Differentiating Fractional Powers

Find the derivative of .

### Answer

Using the definition of the derivative with , we have

Since this limit is tending to , we can use the algebraic technique of multiplying the numerator and denominator by the conjugate of the numerator to manipulate this fraction into a form where we can evaluate the limit. Hence,

Since , we can cancel it from the numerator and denominator as follows:

We can now apply the rules of finite limits to get

Since , we have

The previous example showed us that

We can rewrite this using index notation as which is exactly the formula we would have gotten were we to set in the power rule. This is once again no accident. In fact, not only does the power rule generalize to rational powers, but it also generalizes to any real power.

### Power Rule (General Form)

For any real number ,

We will not prove this most general form of the power rule here. To prove this rigorously, we need to careful define what we mean by when is irrational. However, this is beyond the scope of this explainer.

In the last couple of examples, we will apply the power rule and the sum, difference, and constant multiple rules to find derivatives of more general functions.

### Example 6: Using the Power Rule to Differentiate

Given that , find .

### Answer

Using properties of derivatives, we can differentiate each term independently as follows:

We can now apply the power rule, to each term as follows:

### Example 7: Using the Power Rule to Differentiate

Given that , find .

### Answer

Before we attempt to differentiate this function, we should simplify the expression. Since the denominator consists of a single term, we can use the rules of powers to simplify the expression as follows:

We can now use the power rule, to differentiate each term as follows:

### Key Points

- The power rule for differentiation states that, for any real number ,
- The sum and difference rule states that, for two differentiable functions and ,
- The constant multiple rule states that, for a constant ,
- Using these rules of differentiation, it is possible to find the derivatives of a wide class of functions.