Differential, Differentiation, Derivative

• Derivative

 

- Instantaneous rate of change.

 

- The rate of change of A with respect to B(B에 대한 A의 변화량) :

$\frac{\mathrm{The\; rate\; of\; change\; of\; A}}{\mathrm{The\; rate\; of\; change\; of\; B}}=\frac{dA}{dB}$

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- The left : the slope is 2, that is, dy/dx=2. The rate of change is always constant. This means that the amount of change in y is twice the amount of change in x. In other words, when x increases by 1, y increases by 2, and when x increases by 5, y increases by 10.

 

- The right : The rate of change always changes. 

 

$\text{Average rate of change}=\frac{\mathrm{y\; rate\; of\; change}}{\mathrm{x\; rate\; of\; change}}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{f(b)-f(a)}{b-a}$

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In this figure,

when x = a, the instantaneous rate of change f'(a) is

 

$f^{\mathrm{\prime }}(a)={\left.\frac{dy}{dx}\right|}_{x=a}=\underset{\mathrm{\Delta }x\to 0}{lim} \frac{f(a+\mathrm{\Delta }x)-f(a)}{(a+\mathrm{\Delta }x)-a}$

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when x = x itself, without limiting to a particular value, the instantaneous rate of change f'(x) is

 

$f^{\mathrm{\prime }}(x)=\frac{dy}{dx}=\underset{\mathrm{\Delta }x\to 0}{lim} \frac{f(x+\mathrm{\Delta }x)-f(x)}{(x+\mathrm{\Delta }x)-x}$

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• Slope(Gradient of a line)

 

A line is a number that describes both the direction and the steepness of the line.

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422

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• Polynomial Function Derivative

 

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Using the Factorization above, derive a derivative for the polynomial function below.

 

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• Product Rule 

 

 

https://darkpgmr.tistory.com/45

https://m.blog.naver.com/heoyoonhomath/223007374901

https://blog.naver.com/algosn/221253539965