Differentiating functions: Find the error | Derivative rules | AP Calculus AB | Khan Academy

We're going to do in this video is look at the work of other people as they try to take derivatives and see if their reasoning is correct, and if it's not correct, try to identify what they should have done or where their reasoning went wrong.

So over here it says Nate tried to find the derivative of x^2 + 5x times sin of x. Here is his work. Is Nate's work correct? If not, what's his mistake? So pause the video and see if you can answer this. Is Nate's work correct, and if not, what's his mistake?

All right, so I'm assuming you've had a go at it, so let's work through this step by step. Over here he's just trying to apply the derivative operator to the expression, which is exactly what he needed to do. He's trying to find the derivative of this thing, and he says, "Okay, this is a product of two expressions," and then he says, "Okay, well this is going to be the same thing as the derivative, or this is the same thing as the product of the derivatives."

Now this is a problem you are probably familiar with. If I take the derivative of the sum of two things, so the derivative with respect to X of f(x) + g(x), that indeed is equal to the derivative of the first, f'(x), plus the derivative of the second. But that is not true if we are dealing with the product of functions. The derivative with respect to X of f(x)g(x) is not necessarily - maybe there's some very special circumstances - but in general it's not going to be just the product of the derivatives. It's not going to be just f'(x)g'(x). Here we would want to apply the product rule.

This is going to be equal to the derivative of the first function times the second function plus the first function not taking its derivative times the derivative of the second function. So he should have applied the product rule here, and so let's do that just to see what his answer should have been. I'll get my correcting red pen out here. Say no, that's not what he should have done. He says let's take the derivative of this first thing.

So actually, let me do it in color code, so the derivative of this is 2x + 5. So it should have been (2x + 5) times the second thing, so times sin(x) and then to that he would add the first thing, which is x^2 + 5x, times the derivative of the second thing. So the derivative of sin(x) is cosine(x).

So this is what he should have been seeing at this step right over here. He shouldn't have just taken the product of the derivatives; he should have applied the product rule. So his work is not correct, and his mistake is that he didn't apply the product rule. He just assumed that the derivative of the product is the same thing as the product of the derivatives.

Let's do more examples.

Okay, so let's see. It says Katie tried to find the derivative of (2x^2 - 4) to the third power. Here’s her work. Is Katie's work correct? If not, what is her mistake? So once again, pause the video and see if you can figure it out.

All right, now let’s inspect Katie's work. So she’s taking the derivative of this and let's see over here, looks like she’s taking the derivative of the entire expression with respect to the inner expression. That is close to applying the chain rule properly, but it’s not applying the chain rule properly. So her work is not correct, and her mistake is she’s not correctly applying the chain rule.

Just as a review, the chain rule says look, if we're trying to take the derivative with respect to X of f(g(x)), that this is going to be equal to the derivative of the whole thing with respect to g(x) times the derivative of the inner function with respect to X. So over here we could view our f function as a thing that takes its input and raises it to the third power. So this right over here is f'(g(x)).

But she forgot to multiply it by the derivative of the inner function with respect to X. So she forgot to multiply this by the derivative of (2x^2 - 4) with respect to X, which is going to be, let's see, the derivative of 2x^2 – the power rule, 2 * 2 is 4, so it’s going to be 4x. The derivative of 4 is just zero, so it’s going to be times 4x.

So that’s what she needed to do in order for it to be correct. So she had to have this times 4x here, times 4x. So not correct. She didn’t correctly apply the chain rule.

So let’s do another one of these.

So here it says Nan tried to find the derivative of sin(7x^2 + 4x). Here is his work. Is Nan's work correct? If not, what is his mistake? Pause the video, see if you can figure it out.

All right, so it’s a derivative of sin of this expression, so you’d want to use the chain rule. And finding using the chain rule, you want to find the derivative of the outside function with respect to the inside. So the derivative of sin(something) with respect to that something is going to be cosine of that something. So that’s right, that’s right.

And then you want to multiply that times the derivative of the inside with respect to X. So the derivative of 7x^2 is 14x, and the derivative of 4x is 4. So this is actually - that step looks good. But then Nan does something strange over here. This is the cosine of (7x^2 + 4x), and then that whole thing times (14x + 4).

But they get confused looking at these parentheses, and this tends to happen sometimes. This is actually one of these key errors that the folks at the College Board, the AP folks, told us about. When dealing with these transcendental functions cos, sin, tangent, natural log that are written like this and people see the parentheses and see another parenthesis, their brain just says, “Oh, let me multiply these two expressions in parentheses.” But that’s not right because if we were to add parentheses, this is what this is implying.

So you can’t just take the (14x + 4) and multiply it by this and assume you’re taking the cosine of the whole thing. So this is where Nan makes the mistake. The work is not correct, and the mistake is trying to multiply these two expressions and taking the cosine of the whole thing.

Let’s do one more of these. I find these strangely fun.

All right, this one involves Tom. Tom tried to find the derivative of the square root of (x)/(x^4). Here is his work. Is Tom's work correct? If not, what's his mistake? Pause the video and see if you can figure that out.

So it looks like he’s trying to apply the quotient rule. So applying the quotient rule, you would in the numerator take the derivative of the first expression times the second expression and then minus the first expression times the derivative of the second expression, all of that over the denominator expression squared.

So this looks correct. Actually, it’s a correct application of the quotient rule. It looks like Tom is correctly simplifying. The derivative of x^(1/2) is (1/2)x^(-1/2), so that looks right. The derivative of x^4 is 4x^3, so that looks right. All of this looks algebraically right.

And let’s see, when you simplify this - so let's see, x^(-1/2) * x^4 is indeed x^(4 - 1/2), which looks correct. And that simplifies to that, which looks correct, we’re just using exponent properties there, and then divided everything by - let’s see, oh then everything is in terms of x^(3.5). So we’re going to have -3.5x^(3.5), and then you use exponent properties.

So actually, it looks like he did everything correctly. This is the right answer now. So his work is correct; he did not make any mistakes. But I do have a bone to pick, so to speak, with Tom because he didn’t have to apply the quotient rule here. He did all of this hairy calculus and algebra, but there could have been a very simple simplification he could have made up here.

And this is a key thing to realize. He could have said, "Hey, you know what? This is the same thing as the derivative with respect to X of x^(1/2) * x^(-4)." That’s what the square root of x is and 1 over x^4 is. And so let me color code it.

So that is the same thing as that, and that is the same thing as that. And you wouldn’t even have to use the product rule here; you could simplify this even further. This is the same thing as the derivative with respect to X of just - we have the same base; we can add the exponents. So it’s going to be x^(-3.5).

So you can just use the power rule. So this is going to be equal to bring the -3.5 out front: -3.5x^(-4.5). And then we just decrement this one by one, subtract one from that 4.5 power. So as you can see, he could have gotten this answer much, much, much, much, much, much quicker. But he didn’t make any mistakes; there’s a little bit of a judgment error just immediately going forth with the quotient rule, which gets quite hairy quite fast.