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Course: Differential Calculus>Unit 3

Lesson 9: Second derivatives

Second derivatives

We dive into the fascinating realm of second derivatives, starting with the function y=6/x². Together, we apply the power rule to find the first derivative, then repeat the process to reveal the second derivative. This journey illuminates how we can use mathematical tools to uncover deeper layers of a function's behavior.

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• We've learned so far that a derivative is the slope of a point at a graph; so does the second derivative represent something too?
• The second derivative is the rate of change of the rate of change of a point at a graph (the "slope of the slope" if you will). This can be used to find the acceleration of an object (velocity is given by first derivative). You will later learn about concavity probably and the Second Derivative Test which makes use of the second derivative.
• I get that you aren't actually multiplying when you are finding the second derivative, but why is it the in the denominator of d^2/dx^2 the x gets the second power? why not the d or why not both?
• I'm going to take a shot in the dark for sake of putting this out there: maybe it's because d approaches zero, and is the independent variable? Another thing I'll throw out there is if you nest two limits, the denominator ends up being squared. But I think it's most helpful to just realize that we generally don't pay attention why a radical looks like a radical, so we probably shouldn't put that much effort into critiquing the shape of the second derivative operator either.
• If I calculate the derivative of the second derivative, do I get the "third derivative"? Does this notion exist?
• Yes, you get the third derivative. You can differentiate a (typical, well-behaved) function as many times as you want to get a fourth derivative, fifth derivative, or any nth derivative.
• Simple enough. Question is, what is the use of second derivatives? I know they probably have this-world applications that make them useful, but a priori, this just looks like an unecessary iteration of ... what? Rate of change of a rate of change of a rate of change...?
• Yes, you said it! Rate of change. A classic example for second derivatives is found in basic physics. We know that if we have a position function and take the derivative of this function we get the rate of change, thus the velocity. Now, if we take the derivative of the velocity function we get the acceleration (the second derivative).

Knowing the acceleration is crucially important for various physics applications. Thus, the second derivative is very useful. Many examples like this exist in various disciplines, it is highly important.
• At , Sal says to apply the power rule. I applied the quotient rule, but we both got different answers. Why is that?
• y = 6/x²

Quotient rule method. d/dx [u(x)/v(x)] = [u'(x)•v(x) - u(x)•v'(x)] / [v(x)]²
But I remember it this way (it's equivalent, just rearranging the order) d/dx [u(x)/v(x)] = [v(x)•u'(x) - u(x)•v'(x)] / [v(x)]² so you will my calculation is in the order of the second formula.

dy/dx (6/x²)
= [(x²•0) - (6•2x)] / (x²)²
= (0 -12x) / x⁴
= -12x / x⁴
= -12 / x³ (this is equivalent to -12x‾³ that Sal has)

d²y/dx² (-12/x³)
= [(x³•0) - (-12•3x²)] / (x³)²
= [0 - (-36x²)] / x⁶
= 36x² / x⁶
= 36 / x⁴ (this is equivalent to 36x‾⁴)

Is this what you have?
• What does the second derivative of a function mean?
• If you know that the first derivative of a function is the slope of the original function, the second derivative is the slope of the first derivative.

For instance, the first derivative of a distance versus time graph gives you velocity, whereas the second derivative gives you the acceleration.
• Just a thing I noticed.
If we have a polynomial where all the powers are greater than 0(for example - x^3) if we keep taking the derivative then the function will eventually die(become 0).
But, if instead, we have a polynomial where the powers are less than 0(for example - 1/x) if then we do the same pointless derivations the function will never die it will always be something.
1/x
-1/x^2
2/x^3
-6/x^4
and so on.
• All rational expressions with negative and/or non-integer powers will have infinitely many non-trivial derivatives. (I don't mean to argue over semantics, but an expression with negative powers is not a polynomial, it is rational.)
• Are there any videos that deal with taking higher derivatives implicitly? I just cannot figure out how to get to the second derivative using the first derivative (that I derived implicitly).
• I don't see anything on here either, but to describe it quickly:
Take the derivative of the first derivative, the same way you did the first time, where y'=(dy/dx).
After you finish, replace any dy/dx's with what your answer was for the first derivative, and then simplify.
• In one of the exercise problems for this vides, to get the first derivative of (10/3x^3), the answer says to take (10/3)*d/dx(1/x^3). Why is it not (10)*d/dx (1/3x^3)? This lead me to a different answer of (-10/9x^4) rather than (-10/x^4)
• I think you may have made a mistake, I will go through both variations step by step.

10 d/dx (1/3x^3)
10 d/dx (3x^3)^-1
10 (-(3x^3)^-2*9x^2)
10(-9x^2/(9x^6))
-10/x^4

10/3 d/dx 1/x^3
10/3 d/dx x^-3
10/3 (-3x^-4)
-10x^-4
-10/x^4

Let me know if something didn't make sense. With the first one it looks like you may have had a problem with the chain rule, but if that's not the case again feel free to respond.