So we now have an idea on where to start differentiating (that is, take the derivative of) things like:

```
$\begin{aligned}
4\log(x) \\
x^2 + \log(x) \\
x^2\log(x) \\
\end{aligned}$
```

Our final step is to understand how to differentiate specific items themselves, like *log(x)* and *x ^{2}*.

Let’s start off with how to differentiate *polynomials*. As a reminder, these are expressions that include the following operations:

- addition
- subtraction
- multiplication
- nonnegative exponents

For example:

`$3x^4-2x^2+4$`

To differentiate polynomials, we use the *power rule*. This states the following:

`$\frac{d}{dx}x^{n} = nx^{n-1}$`

To take the derivative of *3x ^{4}*, we do the following:

`$\frac{d}{dx}3x^{4} = 4*3x^{4-1}=12x^{3}$`

Let’s try a challenging problem. Find the derivative of the following polynomial:

`$4x^5+2x$`

Hint: use a concept we learned in the previous exercise. Check your solution below.

## Solution

There are rules even beyond the power rule. Many common functions have defined derivatives. Here are some common ones:

```
$\begin{aligned}
\frac{d}{dx}ln(x) = \frac{1}{x} \\
\frac{d}{dx}e^x = e^x \\
\frac{d}{dx}sin(x) = cos(x) \\
\frac{d}{dx}cos(x) = -sin(x)
\end{aligned}$
```

Let’s use these rules to try out some practice problems!

Practice Problem 1:

`$2x+ln(x)+cos(x)$`

Click below to check your solution.

## Solution

Practice Problem 2:

`$2x*ln(x)$`

Click below to check your solution.

## Solution

Taking derivatives is something that takes plenty of practice to get the hang of. As a learner in this course, your biggest takeaway should be that we can differentiate many common functions by using some combination of these rules in tandem.

### Instructions

The applet to the right gives you more of an idea of what a function and its derivative look like. The functions show are:

```
$\begin{aligned}
f(x)=x^3 \\
f'(x) = 3x^2
\end{aligned}$
```

Using the slider, you can move along different values of both functions. Notice how the tangent line is almost always positive, so the derivative function is completely above the *x*-axis (except at *x=0* when the function’s derivative is equal to zero).