AP®︎/College Calculus BC
Let's explore how to find the derivative of any polynomial using the power rule and additional properties. The derivative of a constant is always 0, and we can pull out a scalar constant when taking the derivative. Furthermore, the derivative of a sum of two functions is simply the sum of their derivatives. Created by Sal Khan.
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- Surely, at0:30, the rule does also works if n = 0, since the derivative will equal 0.x^-1, which equals 0, and that is the derivative of a constant.(114 votes)
- It's pretty nitpicky, but
0 * x ^ (-1)is not always equal to 0. When x equals 0, it's undefined.(152 votes)
- Is there a video where the d/dx notation is explicitly explained? It seems like he justs starts using it from here onward without really talking about it.(30 votes)
- d/dx is just like a operator of differentiation. d(y)/dx will mean taking the derivative of y with respect to x.
The d is for delta or difference so basically it means a change in y with a change in x which gives the derivative or the instantaneous slope at a point.(24 votes)
- Why any number raised to zero power becomes 1?(6 votes)
- For all numbers x (not zero) and all numbers m and n ,
------ = x^(m-n)
This simply means when you are dividing, and the bases are the same, you SUBTRACT the exponents.
------ = 3^(1-1) = 3^0 ; but 3/3 = 1 then we conclude that 3^0 = 1
But isn't only the number 3. All the numbers, that are different from zero, raised to power 0 are equal to one.(7 votes)
- What is a second derivative...is it like an antiderivative(3 votes)
- No, the second derivative is the derivative of the first derivative of any function f(x). It is the change of the rate of change, essentially. The antiderivative, on the other hand, is going backwards from the derivative to the original function. (Later, in calculus, the more common name for the antiderivative becomes the indefinite integral - see integration videos) It is not the same thing as the second derivative.(23 votes)
- What about:
Does that become:
[d/dx f(x)][d/dx g(x)](4 votes)
- No, you will see in a further video the "Product Rule" (https://www.khanacademy.org/math/differential-calculus/taking-derivatives/product_rule/v/applying-the-product-rule-for-derivatives), that can be summed as:
d/dx [f(x)·g(x)] = g(x)·df/dx + f(x)·dg/dx(10 votes)
- Why we can take the constant out directly for the derivative of a constant times the function?(7 votes)
- The product rule is a little bit more than you need for showing this kind of thing. Suppose you've got a function f(x) (and its derivative) in mind and you want to find the derivative of the function g(x) = 2f(x). By the definition of a derivative this is the limit as h goes to 0 of:
(g(x+h) - g(x))/h = (2f(x+h) - 2f(x))/h = 2(f(x+h) - f(x))/h
Now remember that we can take a constant multiple out of a limit, so this could be thought of as 2 times the limit as h goes to 0 of
(f(x+h) - f(x))/h
Which is just 2 times f'(x) (again, by definition). The principle is known as the linearity of the derivative.(4 votes)
- I don't understand why the derivative of a constant is 0. If f(x) = constant, then slope of f(x) = 0. But then, wouldn't the slope of the tangent(the derivative) be different?(4 votes)
- What's going on here is that the tangent line to any line is just the same line again. Think of it this way, the tangent line at x=a should be the line (out of all possible lines) that approximates f(x) the best near x=a. The line that best approximates f(x) = c is the same line again.(4 votes)
- For the example at the very end of the video, the derivative starts with 6x^2, does that mean that the tangent of the function is a curve? How does that work?(3 votes)
- First, remember that the derivative of a function is the slope of the tangent line to the function at any given point.
If you graph the derivative of the function, it would be a curve. Remember though, that this is not the tangent line to the curve, it is only a graph of the derivative, or the slope of the tangent line to the curve at a given point. You can use this graph to find the derivative at a certain point.
For example, let's look at only the first term in the last example in the video, and its derivative. The term is 2x³, and its derivative is 6x².
The graph of 2x³ will look similar to the graph of x³, an odd function moving from the third quadrant towards the first quadrant. The graph of 6x² will look like the typical graph of a quadratic function, which is some variation of a parabola.
Now, let's try to imagine the tangent line to 2x³ at x = 0. Is the tangent line a parabola? No, that's not a line. So, how can we even find a tangent line from the derivative? Finding the value of the derivative at the x-value, and using that as the tangent line's slope. (After all, the derivative is commonly defined as the slope of the tangent line to the function at that x-value.)
At x = 0, the value of 6x² is 0. Thus, the tangent line is a line with slope 0, or a flat line along y = 0 (the value of x³ evaluated at x = 0).
Furthermore, the derivative is a curve because the slope of the tangent line to the function is changing. Think of this as the function increasing or decreasing faster in some intervals, and not so much in others. At x = 0, the derivative is 0. At x = 0.5, x³ is beginning to increase faster, and the derivative is 1.5. At x = 1, the derivative is 6. At x = 2, the derivative is 24. The derivative is clearly not changing at a constant rate with x.(5 votes)
- So can you take a "subset" derivative of a function? for example if the function is to the third power can you take the first derivative (second power) and then take the derivative of that (first power) and then take the derivative of that (0th power) and still call it a derivative of f(x)?(4 votes)
- well, it's the nth derivative of f(x), but yeah it's still a derivative. it's just that deriving zeroes isnt very meaningful(2 votes)
- what does happen when you do 0^0?0:45(3 votes)
- If you're taking the limit of an expression, you should use L'Hôpital's Rule. It only applies when both the numerator and denominator is 0, or infinity. (So 0 divided by infinity or vice versa does not apply)
Here's the link to a more detailed explanation.(4 votes)
Now that we know the power rule, and we saw that in the last video, that the derivative with respect to x, of x to the n, is going to be equal to n times x to the n minus 1 for n not equal 0. I thought I would expose you to a few more rules or concepts or properties of derivatives that essentially will allow us to take the derivative of any polynomial. So this is powerful stuff going on. So the first thing I want to think about is, why this little special case for n not equaling 0? What happens if n equals 0? So let's just think of the situation. Let's try to take the derivative with respect to x of x to the 0 power. Well, what is x to the 0 power going to be? And we can assume that x for this case right over here is not equal to 0. 0 to the 0, weird things happen at that point. But if x does not equal 0, what is x to the 0 power going to be? Well, this is the same thing as the derivative with respect to x of 1. x to the 0 power is just going to be 1. And so what is the derivative with respect to x of 1? And to answer that question, I'll just graph it. I'll just graph f of x equals 1 to make it a little bit clearer. So that's my y-axis. This is my x-axis. And let me graph y equals 1, or f of x equals 1. So that's 1 right over there. f of x equals 1 is just a horizontal line. So that right over there is the graph, y is equal to f of x, which is equal to 1. Now, remember the derivative, one way to conceptualize is just the slope of the tangent line at any point. So what is the slope of the tangent line at this point? And actually, what's the slope at every point? Well, this is a line, so the slope doesn't change. It has a constant slope. And it's a completely horizontal line. It has a slope of 0. So the slope at every point over here, slope is going to be equal to 0. So the slope of this line at any point is just going to be equal to 0. And that's actually going to be true for any constant. The derivative, if I had a function, let's say that f of x is equal to 3. Let's say that's y is equal to 3. What's the derivative of y with respect to x going to be equal to? And I'm intentionally showing you all the different ways of the notation for derivatives. So what's the derivative of y with respect to x? It can also be written as y prime. What's that going to be equal to? Well, it's the slope at any given point. And you see that no matter what x you're looking at, the slope here is going to be 0. So it's going to be 0. So it's not just x to the 0. If you take the derivative of any constant, you're going to get 0. So let me write that. Derivative with respect to x of any constant-- so let's say of a where this is just a constant, that's going to be equal to 0. So pretty straightforward idea. Now let's explore a few more properties. Let's say I want to take the derivative with respect to x of-- let's use the same A. Let's say I have some constant times some function. Well, derivatives work out quite well. You can actually take this little scalar multiplier, this little constant, and take it out of the derivative. This is going to be equal to A. I didn't want to do that magenta color. It's going to be equal to A times the derivative of f of x. Let me do that blue color. And the other way to denote the derivative of f of x is to just say that this is the same thing. This is equal to A times this thing right over here is the exact same thing as f prime of x. Now this might all look like really fancy notation, but I think if I gave you an example it might make some sense. So what about if I were to ask you the derivative with respect to x of 2 times x to the fifth power? Well, this property that I just articulated says, well, this is going to be the same thing as 2 times the derivative of x to the fifth, 2 times the derivative with respect to x of x to the fifth. Essentially, I could just take this scalar multiplier and put it in front of the derivative. So this right here, this is the derivative with respect to x of x to the fifth. And we know how to do that using the power rule. This is going to be equal to 2 times-- let me write that. I want to keep it consistent with the colors. This is going to be 2 times the derivative of x to the fifth. Well, the power rule tells us, n is 5. It's going to be 5x to the 5 minus 1 or 5x to the fourth power. So it's going to be 5x to the fourth power, which is going to be equal to 2 times 5 is 10, x to the fourth. So 2x to the fifth, you can literally just say, OK, the power rule tells me derivative of that is 5x to the fourth. 5 times 2 is 10. So that simplifies our life a good bit. We can now, using the power rule and this one property, take the derivative anything that takes the form Ax to the n power. Now let's think about another very useful derivative property. And these don't just apply to the power rule, they apply to any derivative. But they are especially useful for the power rule because it allows us to construct polynomials and take the derivatives of them. But if I were to take the derivative of the sum of two functions-- so the derivative of, let's say one function is f of x and then the other function is g of x. It's lucky for us that this ends up being the same thing as the derivative of f of x plus the derivative of g of x. So this is the same thing as f-- actually, let me use that derivative operator just to make it clear. It's the same thing as the derivative with respect to x of f of x plus the derivative with respect to x of g of x. So we'll put f of x right over here and put g of x right over there. And so with the other notation, we can say this is going to be the same thing. Derivative with respect to x of f of x, we can write as f prime of x. And the derivative with respect to x of g of x, we can write as g prime of x. Now, once again, this might look like kind of fancy notation to you. But when you see an example, it'll make it pretty clear. If I want to take the derivative with respect to x of let's say x to the third power plus x to the negative 4 power, this just tells us that the derivative of the sum is just the sum of the derivatives. So we can take the derivative of this term using the power rule. So it's going to be 3x squared. And to that, we can add the derivative of this thing right over here. So it's going to be plus-- that's a different shade of blue-- and over here is negative 4. So it's plus negative 4 times x to the negative 4 minus 1, or x to the negative 5 power. So we have-- and I could just simplify a little bit. This is going to be equal to 3x squared minus 4x to the negative 5. And so now we have all the tools we need in our toolkit to essentially take the derivative of any polynomial. So let's give ourselves a little practice there. So let's say that I have-- and I'll do it in white. Let's say that f of x is equal to 2x to the third power minus 7x squared plus 3x minus 100. What is f prime of x? What is the derivative of f with respect to x going to be? Well, we can use the properties that we just said. The derivative of this is just going to be 2 times the derivative of x to the third. Derivative of x to the third is going to be 3x squared, so it's just going to be 2 times 3x squared. What's the derivative of negative 7x squared going to be? Well, it's just going to be negative 7 times the derivative of x squared, which is 2x. What is the derivative of 3x going to be? Well, it's just going to be 3 times the derivative of x, or 3 times the derivative of x to the first. The derivative of x to the first is just 1. So this is just going to be plus 3 times-- we could say 1x to the 0-- but that's just 1. And then finally, what's the derivative of a constant going to be? Let me do that in a different color. What's the derivative of a constant going to be? Well, we covered that at the beginning of this video. The derivative of any constant is just going to be 0, so plus 0. And so now we are ready to simplify. The derivative of f is going to be 2 times 3x squared is just 6x squared. Negative 7 times 2x is negative 14x plus 3. And we don't have to write the 0 there. And we're done. We now have all the properties in our tool belt to find the derivative of any polynomial and actually things that even go beyond polynomials.