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The second derivative, f”, is the derivative of the derivative f’. In other words, in order to find a second derivative, **take the derivative of the derivative.** The second derivative is especially useful when it comes to classifying relative extreme values; if you can take the derivative of a function twice you can determine if a graph of your original function is concave up, concave down, or a point of inflection.

In order to find a **third derivative**, take the derivative one more time (take the derivative of the second derivative). In order to find the fourth derivative, take the derivative another time (take the derivative of the third derivative). Essentially, you can keep on going on and on to infinity with taking derivatives — it is possible to find the hundredth or thousandth or millionth derivative. However, in reality (and with the types of equations you are likely to encounter), you’ll likely only be able to take derivatives up to the fifth derivative. After that, you’ll probably end up with a constant — and while second and third derivatives can give you useful information about a function’s behavior, the hundredth derivative does not.

**Sample question 1:** Find the second derivative of 2x^{3}.

Step 1: **Take the derivative:**

f’ 2x^{3} = 6x^{2}

Step 2: **Take the derivative of your answer from Step 1**:

f’ 6x^{2} = 12x

**Sample question 2:** Find the second derivative of 3x^{5} – 5x^{3} + 3

Step 1: **Take the derivative:**

f’ 3x^{5} – 5x^{3} + 3 = 15x^{4} – 15x^{2} = 15x^{2} (x-1)(x+1)

Step 2: **Take the derivative of your answer from Step 1**:

f’ 15x^{2} (x-1)(x+1) = 60x^{3} – 30x = 30x(2x^{2} – 1)

*That’s it!*

**Warning**: You can’t always take the second derivative of a function. For example, the derivative of 5 is 0.

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