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Dmitriy789 [7]
2 years ago
10

Find the derivative of

" align="absmiddle" class="latex-formula"> by 1st principle of derivative.
Mathematics
1 answer:
sladkih [1.3K]2 years ago
6 0

\huge{\color{magenta}{\fbox{\textsf{\textbf{Answer}}}}}

\frak {\huge{ \frac{1}{1 +  {x}^{2} } }}

Step-by-step explanation:

\sf let \: f(x) =  { \tan }^{ - 1} x \\  \\  \sf f(x + h) =  { \tan}^{ - 1} (x + h)

\sf f'(x) =  \frac{f(x+h)  - f(x) }{h}

\sf \implies \lim_{  h \to 0  } \frac{ { \tan }^{ - 1}(x + h) -  { \tan}^{ - 1}x  }{h}  \\  \\  \\  \sf  \implies  \lim_ {h \to 0}    \frac{  { \tan}^{ - 1} \frac{x + h - x}{1 + (x + h)x} }{h}

By using

\sf { \tan}^{ - 1} x -  { \tan}^{ - 1} y   = \\   \sf { \tan}^{ - 1}  \frac{x - y}{1 + xy} formula

\sf  \implies  \large \lim_{h \to0 }   \frac{  { \tan}^{ - 1}  \frac{h}{1 + hx +  {x}^{2} } }{h}  \\  \\  \\  \sf  \implies   \large{\lim_{h \to0}   } \frac{ { \tan}^{ - 1}  \frac{h}{1 + hx +  {x}^{2} } }{ \frac{h}{1 + hx  +  {x}^{2} }  \times (1 + hx +  {x}^{2} )}  \\  \\  \\  \sf  \implies \large  \lim_{h \to0} \frac{ { \tan}^{ - 1} \frac{h}{1 + hx +  {x}^{2} }  }{ \frac{h}{1 + hx +  {x}^{2} } }  +  \lim_{h \to0} \frac{1}{1 + hx +  {x}^{2} }

<u>Now</u><u> </u><u>putting</u><u> </u><u>the</u><u> </u><u>value</u><u> </u><u>of</u><u> </u><u>h</u><u> </u><u>=</u><u> </u><u>0</u>

<u>\sf  \large  \implies 0 +  \frac{1}{1 + 0 +  {x}^{2} }  \\  \\  \\  \purple{ \boxed  { \implies  \frac{1}{1 +  {x}^{2} } }}</u>

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One hundred rats with mothers that were exposed to high levels of tobacco smoke during pregnancy were put through a simple maze.
Paul [167]

Answer:

The 90% confidence interval for pis (0.7342, 0.8658).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of \pi, and a confidence level of 1-\alpha, we have the following confidence interval of proportions.

\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}

In which

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For this problem, we have that:

100 rats mothers, 80 went right. So n = 100, \pi = \frac{80}{100} = 0.8

90% confidence level

So \alpha = 0.1, z is the value of Z that has a pvalue of 1 - \frac{0.1}{2} = 0.95, so Z = 1.645.

The lower limit of this interval is:

\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.8 - 1.645\sqrt{\frac{0.8*0.2}{100}} = 0.7342

The upper limit of this interval is:

\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.8 + 1.645\sqrt{\frac{0.8*0.2}{100}} = 0.8658

The 90% confidence interval for pis (0.7342, 0.8658).

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3 years ago
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gavmur [86]
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You are offered an item of $17.85 if the sellers states he normally sets the same item for $19.95 what percentage discount it he
nadezda [96]

Answer:

Discount percentage = 10.5% (Approx)

Therefore the discount is seller offering you is 10.5 % .

Step-by-step explanation:

Discount = Marked price - Selling price

As given

You are offered an item for $17.85. If the seller states he normally sells the same item for $19.95 .

Here

Marked price = $19.95

Selling price = $17.85

Discount = 19.95 - 17.85

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Discount percentage = 10.5% (Approx)

Therefore the discount is seller offering you is 10.5 % .

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Step-by-step explanation:

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If you think about it, the question is asking us to find the greatest common factor, or GCF, of the two numbers, 24 and 18. 
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