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$$\text { Find } \frac{d^{2} y}{d x^{2}}, \text { if } 3 x^{2}+2 y^{2}=6$$

$$\frac{-9}{2 y^{3}}$$

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.

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$$\text { Find } \frac{d^{…

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Find $d^{2} y / d x^{2}$.<…

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Calculate $\frac{d^{2} y}{…

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Find $y^{\prime}$ and $y^{…

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02:29

Find $\frac{d^{2} y}{d x^{…

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FInd $d y / d x$$$x^{2…

Find $d^{3} y / d x^{3}$.<…

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Find $d^{2} y / d x^{2}$ i…

question 11 would like you to find d squared y over DX squared if three X squared plus two y squared is equal to six using implicit differentiation. Taking the first derivative with respect to X that would be six x multiplied by DX over D X plus for Y dy DX is equal to zero isolating dy DX that is equal to negative three x over two y. Now we will be taking the driver again with respect to x o d squared y over DX squared. Ah, using the quotient rule here, we can say that this is negative to y times three d x over dx minus three x times two dy dx all over for y squared This dy DX we can sub in here so d squared. Why do you x squared is equal to negative six y minus six x times Negative three x over two y divided by four y squared. That's equivalent to negative six y plus nine x squared over y divided by four. Why squared? Ah, making this a common denominator and also taking out that three so negative three two I squared over Y plus three x squared over why is equal to four y squared? If you combine the denominator in the numerator, your numerator would actually be the same expression in the front. So we can. 76 which means we have negative three times six over why divided by four y squared becomes negative. 18 over for Why cubed? And that is finally equivalent to negative 9/2 y cubed. And that is your final answer to question 12.

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