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Each side of a square is increasing at a rate of $ 6 cm/s. $ At what rate is the area of the square increasing when the area of the square is 16 $ cm^2? $
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Calculus 1 / AB
Harvey Mudd College
University of Michigan - Ann Arbor
University of Nottingham
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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And this question we're asked to find the following, we know that each side of the square is increasing at some rate. And we want to find the rate of the area increase when the area is some area. Well, let's start, wow, let's start first with the area of a square. Remember that the area of a square is equal to just the sides squared? But remember that this is actually a fun. Remember that both the area and the sides, our functions of time now. Uh huh. Yeah. Okay, so it's more correct to say that we have the area as a function of time is equal to the length of the side as a function of time squared. In order to do this problem, we need to differentiate both sides with respect to time and we need to apply the general, remember that the derivative of the area is equal to just the rate of change of the area. So that would just be a prime of T. Equals. And then the derivative of the derivative of the side of the right hand side Is going to be two times S times as prime of T. Now we're given a few things were given that the rate At which the side of the square increases is six cm. So we can we can say safely that s prime of T Is equal to six. Yeah. And we know that the area is 16. If the area is 16, the only way we can get that is this is if S is for Because four square to 16. Sure. So we can plug this information into our equation. So we're going to be having two times S. F. T, which is four times S. Prime F. D. Which is six. And we know that that's going to be equal to 48 and we want centimetres squared per second. And that's how you do this question.
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