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boyakko [2]
3 years ago
11

What is the value of d/dx(1/x) at x=6 ?​

Mathematics
1 answer:
sladkih [1.3K]3 years ago
4 0

If you know your derivative rules, then

d/d<em>x</em> [1/<em>x</em>] = -1/<em>x</em> ²

so that when <em>x</em> = 6, the derivative has a value of -1/36.

If you have to use the definition of the derivative, then

d/d<em>x</em> [1/<em>x</em>] = lim {<em>h</em> → 0} (1/(<em>x</em> + <em>h</em>) - 1/<em>x</em>) / <em>h</em>

… = lim {<em>h</em> → 0} (<em>x</em> - (<em>x</em> + <em>h</em>)) / (<em>hx</em> (<em>x</em> + <em>h</em>))

… = lim {<em>h</em> → 0} (-<em>h</em>) / (<em>hx</em> (<em>x</em> + <em>h</em>))

… = lim {<em>h</em> → 0} (-1) / (<em>x</em> (<em>x</em> + <em>h</em>))

… = -1/<em>x</em> ²

and at <em>x</em> = 6, you again get -1/36.

Alternatively, use the definition of the derivative at a point:

d/d<em>x</em> [1/<em>x</em>] (6) = lim {<em>x</em> → 6} (1/<em>x</em> - 1/6) / (<em>x</em> - 6)

… = lim {<em>x</em> → 6} ((6 - <em>x</em>) / (6<em>x</em>)) / (<em>x</em> - 6)

… = lim {<em>x</em> → 6} -(<em>x</em> - 6) / (6<em>x</em> (<em>x</em> - 6))

… = lim {<em>x</em> → 6} (-1) / (6<em>x</em>)

… = -1/36

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z=\frac{25-24}{\frac{2}{\sqrt{16}}}=2    

p_v =2*P(Z>2)=0.0455  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean differs from 24 at 5% of significance

Step-by-step explanation:

Data given and notation  

\bar X=25 represent the sample mean

\sigma=2 represent the sample population deviation for the sample  

n=16 sample size  

\mu_o =24 represent the value that we want to test

\alpha=0.05 represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

p_v represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the true mean is different from 24, the system of hypothesis would be:  

Null hypothesis:\mu = 24  

Alternative hypothesis:\mu \neq 24  

If we analyze the size for the sample is < 30 but we know the population deviation so is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:  

z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}  (1)  

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

We can replace in formula (1) the info given like this:  

z=\frac{25-24}{\frac{2}{\sqrt{16}}}=2    

P-value

Since is a two sided test the p value would be:  

p_v =2*P(Z>2)=0.0455  

Conclusion  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean differs from 24 at 5% of significance

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