Differentiating Constant Multipliers
Differentiating Polynomials
UBC Calculus Online Course Notes

Differentiating Powers

On this page, we will introduce the


Power Rule

\[ 
  \frac{d}{dx} x^n = nx^{n-1} \mbox{ when } n \mbox{ is a positive integer.} 
 \]



In fact, we have already seen an easy example of this when $  n = 1;  $ that is, we saw that $  \frac{dx}{dx} = 1.  $

To see how the power rule works in another case, let's consider $  \frac{d}{dx}x^2  $ by applying the product rule.

Notice that


\[  \frac{d}{dx}x^2 = \frac{d}{dx}(x \cdot x) = \frac{dx}{dx} x + 
x \frac{dx}{dx} = x + x = 2x 
 \]

which agrees with the power rule, as stated above, when $  n= 2.  $


In the same way, we can compute that


\[ 
 \frac{d}{dx}x^3 = \frac{d}{dx} (x^2\cdot x) = \frac{dx^2}{dx} x + 
x^2 \frac{dx}{dx} = 2x\cdot x + x^2 = 3x^2 
 \]

which again agrees with the power rule.

To see the general version of the power rule, we will apply the process of mathematical induction. That means, we will assume that we have verified the power rule for values up to some number $  N 
 $ (for instance, we have already verified it above for values up to $  N = 3  $ ). We will then show that we can verify the power rule for the next integer $  N+1. $ This means that if we know the power rule holds for some exponent, it also holds for the next one. Since we know that it holds for $  n=3,  $ it must also hold for $  n=4.  $ Then since we know it holds for $  n=4, 
 $ we also know it holds for $  n=5.  $ Now you see the picture: it also must hold for $  n=6,7,8,\ldots 
 $


Now we will assume that we have verified the power rule up to some exponent $  N.  $ Let's consider the next exponent:

\[   \frac{d}{dx}(x^{N+1}) = \frac{d}{dx}(x^N\cdot x) = 
\frac{dx^N}{dx} x + x^N\frac{dx}{dx} = Nx^N + x^N = (N+1)x^N. 
 \]


This means that the power rule has been verified for the next power and so it holds for every power.


Note: If this is confusing to you, you may want to look in a text book for another way of understanding the Power Rule.



Example:

The Power Rule tells us that the derivative $ 
   \frac{d}{dx}(x^5) = 5x^4. 
 $

Below, the graph of the function $  y(x) = x^5 
 $ is shown on the left while its derivative $  5x^4 
 $ is shown on the right.



Questions:

  1. Can you explain the relationship between these two graphs?

  2. What can you infer from the fact that the derivative is always positive?

  3. What happens at the point where the derivative is equal to zero?