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 x².

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 3x⁴, 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.