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### Course: Calculus 1>Unit 3

Lesson 9: Second derivatives

# Second derivatives (implicit equations): evaluate derivative

Given the first derivative of an implicit equation in x and y, evaluate the second derivative at a certain point. Problem taken from the 2015 AP Calculus AB exam.

## Want to join the conversation?

• How is it apparent that you should use the quotient rule here, and not rewrite the expression to use the product rule?
• The denominator here is complex, so the denominator to the (-1) would be much more complicated, so we just use the quotient rule and then substitute the value of x,y respectively.
• Even if the method is straightforward, the calculation involved is very challenging.
• Why do we put the squared on the d on the top and the squared on the x on the bottom of the derivative operator?
(1 vote)
• dy is called the differential of y; you can think of it as an infinitesimally small change in y. In other approaches to calculus, we define the derivative of y with respect to x as the differential of y divided by the differential of x, dy/dx.

To take the second derivative of y with respect to x, we take the differential of the differential of y, d(dy), and divide it by dx twice. So the second derivative is d(dy)/(dx)².

Then we abbreviate ddy as d²y, and drop the parentheses in the denominator to get d²y/dx².
• Does it matter if you calculate the value of the first derivative (1/4 in this case) and then sub it into the second derivative equation? Or is it equally acceptable to sub the equation for the first derivative directly into the second derivative equation and then use the values of x and y to calculate?
• Did you try them both for the problem shown in this video?

What did you find?

Both of those approaches are mathematically equivalent and should give identical results ...
• This isn't related to the question but why is the second derivative written as d^2y/dx^2?
(1 vote)
• Sal explains it in the first video on Second Derivatives. Basically, d^(2)y/dx^(2) is d/dx(dy/dx). So, the numerator kinda becomes d^(2)y and the denominator becomes dx^(2). Note that you don't multiply the terms, as d^(2)/dx^(2) is an operator just like d/dx. All in all, it's pretty much a convention someone long ago came up with, and we continue using it
• In practice problems for second derivatives, implicit equations, problem 2 (x^3+y^2=24), step 3/5: what manipulation did you do to go from [-6xy+3x^2(-3x^2/2y)]/2y^2 to [-12xy^2+9x^4]/4y^3?
(1 vote)
• So this is equivalent fractions. Remember yr 7 math.

You have just multiplied the numerator and denominator by 2y.
(1 vote)
• When doing the practice examples I have a question. The second derivatives that take a derivative of -(x/y) have a quotient rule that doesn't match a product rule answer, why is that? I'm assuming it has something to do with the problem of having two unknowns. Shouldn't the functions f(x)=x and g(y)=y, if the broader function is f(x)/g(x), shouldn't this still be equivalent to f(x)*g(x)^-1?
(1 vote)
• It is important to be clear. If you really want you could use a derivative calculator.

If you could write your solution then I can check it. I would avoid using quotient rule as it is more complicated then product rule.

To answer though product rule/quotient rule are equivalent
(1 vote)
• I think you have plugged in (1) while replacing x instead of (-1) at , and the correct answer is -3/32. Is this correct?
(1 vote)
• Can someone show this using the product rule? I know it is more tedious but I want to see how it achieves the same result.
So far I have y(3y^2-x)^-1
Then dy/dx(y)*(3y^2-x)+ y*dy/dx(3y^2-x)^-1
Then (1/4)1*4 + 1*dy/dx(3y^2-x)^-1
Then 1+dy/dx(3y^2-x)^-1
Then using the chain rule >> 1+ (dy/dx(-(3y^2-x)^-2* (dy/dx(6y-1))

Mainly where I'm stuck is where to input the dy/dx into the chain rule section. Once I got -5/16, then 1/128. I'm not getting the same result...
(1 vote)
• The fractions get really ugly (and unreadable) so I recommend you render this in LaTeX so it'll look better.

If you have $y(3y^{2} - x)^{-1}$, on differentiating using the product and chain rule, we get $y(-(3y^{2}-x)^{-2})(6y(\frac{dy}{dx})-1) + (3y^{2}-x)^{-1}(\frac{dy}{dx})$. Now, from the already given equation of $\frac{dy}{dx}$, we can find $\frac{dy}{dx} = \frac{1}{4}$. Plus, we have $x = -1$ and $y = 1$. So, on plugging all of this in, we get $(1)(-(3(1)^{2}-(-1))^{-2})(6(1)(\frac{1}{4})-1) + (3(1)^{2}-(-1))^{-1}(\frac{1}{4})$. This simplifies to $(\frac{-1}{16})(\frac{6}{4}-1) + (\frac{1}{4})(\frac{1}{4})$, which is $(\frac{-1}{32}) + (\frac{1}{16})$ which is $\frac{1}{32}$. So, as you can see, the product rule works too!
(1 vote)
• Why can't we simplify it to -y^2?
(1 vote)

## Video transcript

- [Instructor] So we have a question here from the 2015 AP Calculus AB test, and it says consider the curve given by the equation y to the third minus x y is equal to two. It can be shown that the first derivative of y with respect to x is equal to that. So they solve that for us. And then part c of it, I skipped parts a and b for the sake of this video. Evaluate the second derivative of y with respect to x at the point on the curve where x equal negative one and y is equal to one. So pause this video and see if you can do that. All right now let's do it together. And so let me just first write down the first derivative. So dy, derivative of y, with respect to x is equal to y over three y squared minus x. Well if we're concerning ourselves with the second derivative, well then we wanna take the derivative with respect to x of both sides of this. So let's just do that. Do the derivative operator on both sides right over here. Now on the left hand side, we of course are going to get the second derivative of y, with respect to x. But what do we get on the right hand side? And there's multiple ways to approach this. But for something like this, the quotient rule probably is the best way to tackle it. I sometimes complain about the quotient rule saying hey, it's just a variation of the product rule. But it's actually quite useful in something like this. We just have to remind ourselves that this is going to be equal to the derivative of the numerator with respect to x. And so that's just going to be derivative of y with respect to x times the denominator, three y squared minus x Minus the numerator, y, times the derivative of the denominator with respect to x. Well, what's the derivative of this denominator with respect to x? Well the derivative of three y squared with respect to x that's going to be the derivative of three y squared with respect to y, which is just going to be six y. I'm just using the power rule there. Times the derivative of y with respect to x. All I did just now is I took the derivative of that with respect to x. Which is derivative of that with respect to y times the derivative of y with respect to x. Comes straight out of the chain rule. Minus the derivative of this with respect to x, which is just going to be equal to one. All of that over, remember, we're in the middle of the quotient rule right over here. All of that over the denominator squared. All of that over three y squared minus x, squared. Now lucky for us, they want us to evaluate this at a point, as opposed to have to do a bunch of algebraic simplification here. So we can say, when, let me do it over here. So when, I'll it right here. When x is equal to negative one, and y is equal to one, well first of all, what's the, what's dy dx going to be? The derivative of y with respect to x, let me scroll down a little bit, so we have a little bit more space. The derivative of y with respect to x is going to be equal to one over three times one squared, which is just three, minus negative one. So that's just going be plus one. It's going to be equal to 1/4. And so this whole expression over here, so I can write the second derivative of y with respect to x is going to be equal to, well we know that that's going to be equal to 1/4. 1/4 times three times one squared, which is just three, minus negative one, so plus one, minus one, so I'll just leave that minus out there, times six times one, times 1/4. Let me just write it out. Six, six times one times 1/4 minus one all of that over let's see this is going to be three times y squared, y is one, so is just gonna be three, three minus negative one, so plus one, squared. Now what is this going to be? And this is just simplifying something here. 1/4 times four that's going to simplify to one. And let's see, this is going to be 1 1/2 minus one so that's going to be 1/2. And then we're going to be have all of that over 16. And so this is going to be equal to, we get a mini drum roll here, this is going to be equal to one minus 1/2, which is equal to 1/2. Over 16, which is the same thing as one over 32. And we are done.