Second derivatives review
Review your knowledge of second derivatives.
What are second derivatives?
The second derivative of a function is simply the derivative of the function's derivative.
Let's consider, for example, the function . Its first derivative is . To find its second derivative, , we need to differentiate . When we do this, we find that .
Want to learn more about second derivatives? Check out this video.
Notation for second derivatives
We already saw Lagrange's notation for second derivative, .
Leibniz's notation for second derivative is . For example, the Leibniz notation for the second derivative of is .
Check your understanding
Want to try more problems like this? Check out this exercise.
Want to join the conversation?
- Why does the denominator read dx^2 instead of (dx)^2? Wouldn't the latter be the proper notation since the operator d/dx is applied to dy/dx, hence making it d(y)^2/(dx)^2?
If someone could answer with an authentic source to verify it, I would greatly appreciate it. Thanks!(31 votes)
- You're right, it should be (dx)^2. It's just a laziness in notation, no one really wanted to write out the parentheses each time. I don't have a source on this, I just heard it somewhere.(17 votes)
- what does the second derivative tell us about?(6 votes)
- It tells us the rate of change of the rate of change. For example, acceleration is the second derivative of a position function, like velocity is the first derivative.(6 votes)
- why the second derivative operator is not d^2y/((d^2)(x^2)), i think this way is because the product of d/dx and dy/dx is d^2y/((d^2)(x^2))(2 votes)
- It's just a (poor and confusing) convention, but when Leibnitz first invented this notation, he thought of units of physical quantities. For example, the second derivative (d^2y/dx^2) of position is acceleration. Acceleration has the units of m/s^2. And hence, the derivative (excluding the "d" part) is also y/x^2.
There's another thing to consider that dx isn't d times x. It isn't a product and hence, dx * dx can't be d^2x^2 (d is an operator). But, we write d^2 in the numerator anyway, so this kinda invalidates it.
Honestly speaking, this is the best explanation I could find. There's no reason why it couldn't have been d^y/d^2x^2. People used d^2y/dx^2 and we got used to it.(4 votes)
- is there such thing as third derivative?(1 vote)
- Yes, we can find any number of derivatives as long as each derivative is also differentiable.(5 votes)
- In problem #1. Why doesn't the 2 get multiplied against the entire derivative of cos(x/2)? It is only being multiplied by the first part of the chain: -sin(x/2). That doesn't seem correct.(1 vote)
- It was multiplied by the derivative, that is why it went away. The derivative of 2cos(x/2) is 2 d/dx cos(x/2). You can use the chain rule to differentiate it, so you get 2*(1/2*-sin(x/2)). This simplifies to -sin(x/2) because 2 * 1/2 = 1. Does this help?(5 votes)
- Given: dy/dx = x/y. Find the 2nd derivative d2y/dx2 in terms of x and y. I found (y^2 - x^2) / y^3 and it was marked as correct. Why then do certain calculators show the answer as
y/x ? Obviously, the calculator knows something I don't. What is it?(2 votes)
- If I am asked to find f'(x) of f(x)=x^4, does that mean that I am to find the second derivative (or f"(x))?(0 votes)
- If you are asked to find the first derivative (f'(x)), you are to find the first derivative (f'(x)).(0 votes)