Tells me derivative of that is 5x to the fourth. You can literally just say, OK, the power rule The fourth power, which is going to be equal to 2 To the 5 minus 1 or 5x to the fourth power. This is going to be 2 times theĭerivative of x to the fifth. I want to keep itĬonsistent with the colors. This is going to be equal toĢ times- let me write that. Take this scalar multiplier and put it in front To be the same thing as 2 times the derivative of x to theįifth, 2 times the derivative with respect to x That I just articulated says, well, this is going You the derivative with respect to x of 2 times x Gave you an example it might make some sense. Like really fancy notation, but I think if I This thing right over here is the exact same The derivative of f of x is to just say that Little scalar multiplier, this little constant, and The derivative with respect to x of- let's use Is just a constant, that's going to be equal to 0. To x of any constant- so let's say of a where this What x you're looking at, the slope here is going to be 0. So what's the derivativeĪt any given point. You all the different ways of the notation for derivatives. Of y with respect to x going to be equal to? And I'm intentionally showing The derivative, if I had aįunction, let's say that f of x is equal to 3. Line at any point is just going to be equal to 0. Point over here, slope is going to be equal to 0. Slope at every point? Well, this is a line, so Tangent line at this point? And actually, what's the One way to conceptualize is just the slope of the There is the graph, y is equal to f of x, X equals 1 to make it a little bit clearer. With respect to x of 1? And to answer that question, 0 to the 0, weird thingsĮqual 0, what is x to the 0 power going to be? Well, this is the same thingĪs the derivative with respect to x of 1. This case right over here is not equal to 0. Well, what is x to theĠ power going to be? And we can assume that x for Want to think about is, why this little specialĬase for n not equaling 0? What happens if n equals 0? So let's just thinkĭerivative with respect to x of x to the 0 power. That essentially will allow us to take the derivative To a few more rules or concepts or properties of derivatives The derivative with respect to x, of x to the n, is going toīe equal to n times x to the n minus 1 for n not equal 0. Rule, and we saw that in the last video, that The derivative is clearly not changing at a constant rate with x. At x = 0.5, x³ is beginning to increase faster, and the derivative is 1.5. Think of this as the function increasing or decreasing faster in some intervals, and not so much in others. 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).įurthermore, the derivative is a curve because the slope of the tangent line to the function is changing. (After all, the derivative is commonly defined as the slope of the tangent line to the function at that x-value.)Īt x = 0, the value of 6x² is 0. 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. Is the tangent line a parabola? No, that's not a line. Now, let's try to imagine the tangent line to 2x³ at x = 0. The graph of 6x² will look like the typical graph of a quadratic function, which is some variation of a parabola. 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 term is 2x³, and its derivative is 6x². You can use this graph to find the derivative at a certain point.įor example, let's look at only the first term in the last example in the video, and its derivative. 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. If you graph the derivative of the function, it would be a curve. First, remember that the derivative of a function is the slope of the tangent line to the function at any given point.
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