Basic derivative rules: find the error | Derivative rules | AP Calculus AB | Khan Academy

So we have two examples here of someone trying to find the derivative of an expression.

On the left-hand side, it says Avery tried to find the derivative of 7 - 5x using basic differentiation rules; here is her work.

On the right-hand side, it says Hannah tried to find the derivative of -3 + 8x using basic differentiation rules; here is her work.

These are two different examples from the differentiation rules exercise on KH Academy, and I thought I would just do them side by side because we can think about what each of these people are doing, correct or incorrect.

These are similar expressions; we have a constant and then we have a first-degree term, a constant, and then a first-degree term. So they're going to take the derivative.

Let's see. Step one for Avery: she took her separately taking the derivative of seven and separately taking the derivative of 5x. So this is, my Spider Sense is already going off here because what happened to this negative right over here? So it would have made sense for her to do the derivative of 7, and she could have said minus the derivative of 5x.

That's one possibility that she could have done. The derivative of a difference is equal to the difference of the derivatives; we've seen that property. Or she could have said the derivative; she could have said this is equal to the derivative of 7 plus the derivative with respect to X of -5x.

These two things would have been equivalent to this one, but for this one, she somehow forgot to include the negative. So I think she had a problem right at step one.

Now, if you just follow her logic after step one, let's see if she makes any more mistakes. She takes the derivative of a constant. So a constant isn't going to change with respect to X. So that makes sense that that derivative is zero.

And so we still have the derivative of 5x, and remember it should have been -5x or minus the derivative of 5x. Let's see what she does here. So that zero disappears, and now she takes the constant out, and that's true; the derivative of a constant times something is equal to the constant times the derivative of that something.

Then she finds that the derivative with respect to X of X is one, and that's true. The slope, if you had the graph of y = x, the slope there is one, or what's the rate of change at which X changes with respect to X? Well, it's going to be one for one.

So the slope here is one, so this is going to be 5 * 1, which is equal to five. At the end, they just say what step did Avery make a mistake? So she clearly made a mistake at step one.

This right over here should have been a negative. If that's a negative, then that would have been a negative, then this would have been a negative, then that would have been a negative, and then her final answer should have been -5.

Now, let's go back to Hannah to see if she made any mistakes and where. So she's differentiating a similar expression. So first, she takes the derivative of the constant plus the derivative of the first-degree term; the derivative of a constant is zero.

That looks good. So you get the zero, and then you have the derivative of the first-degree term; that's what she's trying to figure out. Then let's see. She's taking... okay, so this seems off. She is assuming that the derivative of a product is equal to the product of the derivatives.

That is not the case, especially if you have a constant here. There's actually a much simpler way of thinking about it. Frankly, the way that Avery thought about it—Avery had made a mistake at step one—but this is actually going to be equal to the derivative of a constant times an expression is equal to the same thing as the constant times the derivative of the expression.

So this would have been the correct way to go. Then the derivative of x with respect to X—well, that's just going to be one. So this should have all simplified to eight.

What she did is she assumed—she tried to take the derivative of eight and multiply that times the derivative of x. That is not the way it works. In the future, you will learn something called the product rule, but you wouldn't even have to apply that here because one of these components, I guess you could say, is a constant.

So this is the wrong step. This is where Hannah makes a mistake. You can see instead of getting a final answer of eight, she is getting a final answer of—she assumes, well, the derivative of 8 is 0 times the derivative of x is 1, 0 * 1, and she gets zero, which is not the right answer.

So she makes a mistake at step three, and Avery made a mistake at step one.