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

4log(x)x2+log(x)x2log(x)\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:

3x42x2+43x^4-2x^2+4

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

ddxxn=nxn1\frac{d}{dx}x^{n} = nx^{n-1}

To take the derivative of 3x4, we do the following:

ddx3x4=43x41=12x3\frac{d}{dx}3x^{4} = 4*3x^{4-1}=12x^{3}

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

4x5+2x4x^5+2x

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

Solution The final answer is 20*x^3+2]

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

ddxln(x)=1xddxex=exddxsin(x)=cos(x)ddxcos(x)=sin(x)\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)2x+ln(x)+cos(x)

Click below to check your solution.

Solution The final answer is 2+1/x-sin(x)]

Practice Problem 2:

2xln(x)2x*ln(x)

Click below to check your solution.

Solution The final answer is 2ln(x)+2]

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:

f(x)=x3f(x)=3x2\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).

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