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.
The second derivative is found by differentiating the derivative. It can be written in two ways – in function notation, the second derivative of f(𝑥) is written as f’’(𝑥). With Leibniz’s notation, the second derivative of 𝑦 with respect to 𝑥 is given by (said “d-two-y by d-x-squared”).
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 𝑦 = 3𝑥5 – 4𝑥2 + 5𝑥 – 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”(𝑥).
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 – we differentiate twice.
You can leave the result in this format or simplify it:
1. Find the second derivative of the following functions:
a. f(𝑥) = 8𝑥2 – 3𝑥 + 4
b. f(𝑥) = 2𝑥3 – 15𝑥
c. 𝑦 = 𝑥4 + 4𝑥3 – 𝑥2
2. Find the second derivative for each of the functions below.
b. 𝑦 = (2𝑥 + 1)(𝑥 – 5)(𝑥 + 1)
3. Given that f(𝑥) = 4𝑥3 – 2𝑥 + 5, find each of the values below.
4. Find the 𝑥-coordinates at which the second derivatives of the following function equals 16.
5. Given that 𝑦 = p𝑥4 – 2p𝑥3 + 3p𝑥2 – 𝑥 + 2, and when 𝑥 = -2, = 39, find the value of 𝑝.
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