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vampirchik [111]
3 years ago
8

How to differentiate y=x^n using the first principle. In this question, I cannot use the rule of differentiation. I have to do t

he old fashioned way. How do I do it?​
Mathematics
1 answer:
Zarrin [17]3 years ago
4 0

By first principles, the derivative is

\displaystyle\lim_{h\to0}\frac{(x+h)^n-x^n}h

Use the binomial theorem to expand the numerator:

(x+h)^n=\displaystyle\sum_{i=0}^n\binom nix^{n-i}h^i=\binom n0x^n+\binom n1x^{n-1}h+\cdots+\binom nnh^n

(x+h)^n=x^n+nx^{n-1}h+\dfrac{n(n-1)}2x^{n-2}h^2+\cdots+nxh^{n-1}+h^n

where

\dbinom nk=\dfrac{n!}{k!(n-k)!}

The first term is eliminated, and the limit is

\displaystyle\lim_{h\to0}\frac{nx^{n-1}h+\dfrac{n(n-1)}2x^{n-2}h^2+\cdots+nxh^{n-1}+h^n}h

A power of h in every term of the numerator cancels with h in the denominator:

\displaystyle\lim_{h\to0}\left(nx^{n-1}+\dfrac{n(n-1)}2x^{n-2}h+\cdots+nxh^{n-2}+h^{n-1}\right)

Finally, each term containing h approaches 0 as h\to0, and the derivative is

y=x^n\implies y'=nx^{n-1}

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8 0
2 years ago
Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for t
rjkz [21]

Answer:

COV (all stocks) = 0.55

COV (stocks and bonds) = 0.82

Step-by-step explanation:

Coefficient of Variation is used to measure variability.

It is defined as the ration of standard deviation and the mean.

It can be used to compare variability of two population or two samples.

Formula:

\text{Coefficient of Variation} = \displaystyle\frac{\text{Standard Deviation}}{\text{Mean}}

\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}

where x_i are data points, \bar{x} is the mean and n is the number of observations.

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x: 14, 0, 39, 25, 32, 27, 28, 14, 14, 15

Mean = \frac{208}{10} = 20.8

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y = 6, 2, 29, 17, 26, 17, 17, 2, 3, 5

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Since coefficient of variation of x is less compared to y, thus it could be said bonds does not reduce overall risk of an investment portfolio.

6 0
3 years ago
A triangular sail has a base length of 2.5 meters.The area of the sail is 3.75 square meters.How tall is the sail?
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Area of a triangle = 1/2 x base x height

--------------------------------------
Find Height :
--------------------------------------
3.75  = 1/2 x 2.5 x height

3.75 = 1.25 x height 

height = 3.75 ÷ 1.25

height = 3 meters

----------------------------------------------------------------------------
Answer: The height is 3 meters.
----------------------------------------------------------------------------
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