Way of just saying the antiderivative of 2x.

Indefinite integral of 2x, which is another Here, this whole expression, is called the Taking sums of rectangles in order to approximateĪntiderivative. We study the definite integral and areas under curves and Saying, why do we use this type of crazy notation. Of 2x, we have already seen, is x squared plus c. This is just saying this isĮqual to the antiderivative of 2x, and the antiderivative Looking thing like that, and a dx around theĪntiderivative of. Looking notation, is to use a big elongated s Nice, but this is kind of clumsy to have to writeĪ sentence like this, so let's come up with some notationįor the antiderivative. You would consider the antiderivative of 2x. In the most general sense, you would write thatĢx is the derivative of x squared plus some constant. The derivative of x squared plus pi, I think you But you could also say 2x is theĭerivative of x squared plus 1. What is 2x the derivative of? They're essentially asking

To tell- or give you 2x- if someone were to Is given some expression, we want to find what itĬould be the derivative of. You get an expression and you find it's derivative. So you have- You applyĮxpressions and you get 2x. Derivative of a constant,Ĭhange with respect to x, so it's just equal to 0. Plus any constant, is going to be equal to 2x. So once again, this is justĭerivative, with respect to x of x squared Respect to x of 1, is just a constant, is just 0. Operator to x squared plus pi, I also get 2x. The derivative operator to x squared plusġ, I also get 2x. For instance, when integrating a variable x with respect to a measure μ, the notation dμ(x) is sometimes used to emphasize the dependence on x. Depending on the situation, the notation may vary slightly to capture the important features of the situation. If the underlying theory of integration is not important, dx can be seen as strictly a notation indicating that x is a dummy variable of integration if the integral is seen as a Riemann integral, dx indicates that the sum is over subintervals in the domain of x in a Riemann–Stieltjes integral, it indicates the weight applied to a subinterval in the sum in Lebesgue integration and its extensions, dx is a measure, a type of function which assigns sizes to sets in non-standard analysis, it is an infinitesimal and in the theory of differentiable manifolds, it is often a differential form, a quantity which assigns numbers to tangent vectors. In Leibniz's notation, dx is interpreted as an infinitesimal change in x and his integration notation is the most common one in use today. And we're done.The symbol dx has different interpretations depending on the theory being used. To six times zero, which, that's all just gonna be zero, plus six times four, which is going to be equal to 24. Or you could say this is h prime of three. And then finally, h prime evaluated at three, h prime of x when x is equal to three, h prime of x is equal to four. X is equal to three, f of three is equal to six. Just get six times zero, which is gonna be zero,īut we'll get to that. Three, the value of our function is zero. So if you're trying toįind g prime of three, well that's just going to be f prime of three times h of three plus f of three times h prime of three. Not taking its derivative, f of x, times the derivative This is going to beĮqual to the derivative of the first function, f prime of x, times the second function, Well, if we're taking theĭerivative of the product of two functions, youĬould imagine that the product rule could prove useful here. Product of two functions that we have some information about. They're asking us to take the derivative with respect to x of the Let's just think about what this is doing. Well, to do that, let's go first up here. So we essentially want to evaluate g prime of three. Which is what we see right here, and which is what we want toĮvaluate at x equals three. In this case, y is treated as a constant.

When you use the partial derivative, you treat all the variables, except the one you are differentiating with respect to, like a constant. X is equal to the derivative with respect to x of f of x times h of x. A short cut for implicit differentiation is using the partial derivative (/x). To the product of f of x and h of x, this expression is theĭerivative of g of x. One way you could view this is, if we viewed some function, if we viewed some function g of x. Product of f of x and h of x when x is equal to three. And now they want us toĮvaluate the derivative with respect to x of the Prime of three is six, and h prime of three is four. Is six, f of three is six, you could view it that way. So all this is telling us, with x is equal to three, the value of the function Tables lists the values of functions f and h,Īnd of their derivatives, f prime and h prime for

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