If you’re looking for revision info on second order derivatives, then you’ve come to the right place. The derivative of a function (if it can be differentiated) is a function that describes the rate of change at any point on the original function. As the derivative is a function, it can also be differentiated, giving the **second derivative**.

If any of these terms aren’t familiar to you, check out our blogs on Differentiation from First Principles and Differentiation of Functions of the Form *x*^{n}.

If you’d like to download this blog in PDF or PowerPoint formats, click here. To practice some of the prior knowledge required for this topic try these multiple-choice questions.

The second derivative is found by differentiating the derivative. It can be written in two ways – in function notation, the second derivative of f(š„) is written as fāā(š„). With Leibnizās notation, the second derivative of š¦ with respect to š„ is given by (said ād-two-y by d-x-squaredā).

Since the second derivative is the derivative of the gradient function, it gives the rate of change of the gradient.

**Example Question 1**

Given that š¦ = 3š„^{5} – 4š„^{2 }+ 5š„ – 3, find .

First, differentiate to get . Remember, multiply by the power then reduce the power by 1.

Now, just differentiate again:

**Example Question 2**

Given that , find f”(š„).

This time, you need to write the function using index notation before differentiating:

This example is almost identical to example 1 but with different notation. Use the same process ā we differentiate twice.

You can leave the result in this format or simplify it:

**Practice Questions**

1. Find the second derivative of the following functions:

a. f(š„) = 8š„^{2} ā 3š„ + 4

b. f(š„) = 2š„^{3 }ā 15š„

c. š¦ = š„^{4} + 4š„^{3} ā š„^{2}

2. Find the second derivative for each of the functions below.

a.

b. š¦ = (2š„ + 1)(š„ – 5)(š„ + 1)

3. Given that f(š„) = 4š„^{3} ā 2š„ + 5, find each of the values below.

a. f(3)

b. f'(3)

c. f”(3)

4. Find the š„-coordinates at which the second derivatives of the following function equals 16.

5. Given that š¦ = pš„^{4} ā 2pš„^{3} + 3pš„^{2} ā š„ + 2, and when š„ = -2, = 39, find the value of š.

**Answers**

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