The Power Rule is a helpful tool in calculus. It makes finding the slopes, or derivatives, of polynomial functions easier.

Here's what the Power Rule says:

If you have a function that looks like this:
f(x) = ax^n

In this case:

• a is a constant (a number that doesn’t change)
• n is a positive whole number (like 1, 2, 3, etc.)

Then, the derivative, or slope, can be found using this formula:
f'(x) = nax^(n-1)

Let’s look at a couple of examples to see how it works!

Example 1:
For the function f(x) = 3x^4, we apply the Power Rule like this:

• First, we take the exponent (4) and multiply it by 3.
• Then, we decrease the exponent by 1.

So, we get:
f'(x) = 4 × 3x^(4-1) = 12x^3

Example 2:
Now, let’s look at another function: g(x) = 5x^2 + 2x.

Using the Power Rule here gives us:

• For the first part, 5x^2, we take 2 times 5 which equals 10, and lower our exponent from 2 to 1.
• For the second part, 2x, we take 1 times 2 and lower the exponent from 1 to 0.

So, we find:
g'(x) = 2 × 5x^(2-1) + 1 × 2x^(1-1) = 10x + 2

In simple terms, using the Power Rule makes it much easier to find derivatives and helps us work with more complicated functions quickly!