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$$\text { Find } \frac{d^{2} y}{d x^{2}}, \text { if } x^{1 / 2}+y^{1 / 2}=6$$

$$3 / x^{3 / 2}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

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Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

01:20

$$\text { Find } \frac{d^{…

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01:54

Find $\frac{d^{2} y}{d x^{…

04:13

Find $d^{2} y / d x^{2}$

04:37

Find $d^{2} y / d x^{2}.$<…

00:41

Find $d^{2} y / d x^{2}$.<…

02:54

01:35

Find $d y / d x$.$$

01:02

02:29

02:39

01:23

FInd $d y / d x$$$x^{2…

question nine is also unifying d squared y over DX squared or the second derivative of X to the one half plus y to the one half equals six. So starting off, you can isolate. Why So why to the one half equal six minus X to the one half? Scoring both sides y equals six minus X to the one half squared. Uh, now taking the derivative of Y and respect to X Do I. D. X is equal to two times six minus X to the one half multiplied by negative one half X to the negative one half because of the chain roll. From there, you can simplify that to negative six minus X to the one half divided by X to the one half. Ah, dividing these individually, you have negative six two divided by X to the one half plus one, which is equivalent to just negative six x to the negative one half plus one. Now taking second derivative of why, with respect to X, do you d squared y over DX squared would just be negative. Six times negative one half X to the negative 3/2, which is equivalent to three over X to the three over to, and that is your final answer for question nine

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