xtan-1x differentiate using first principle
Answer:
[tex]F'(x) = \frac{x}{{1 +x^{2} } } + tan^{-1} (x)}[/tex]
Explanation:
Given - F(x) = xtan⁻¹x
To find - Differentiate using first principle
Formula used -
First Principal :
F'(x) = [tex]\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}[/tex]
and
[tex]tan^{-1}(A - B) = tan^{-1}(\frac{A - B}{1 + AB} )[/tex]
Proof -
Given that, F(x) = xtan⁻¹x
⇒F(x+h) = (x+h) tan⁻¹(x+h)
So,
[tex]F'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\\= \lim_{h \to 0} \frac{(x+h)tan^{-1} (x+h) - xtan^{-1} x}{h}\\= \lim_{h \to 0} \frac{xtan^{-1} (x+h) + htan^{-1} (x+h) - xtan^{-1} x}{h}\\= \lim_{h \to 0} \frac{x(tan^{-1} (x+h) - tan^{-1} x) + htan^{-1} (x+h)}{h}\\= \lim_{h \to 0} \frac{x}{h} (tan^{-1} (x+h) - tan^{-1} x) + tan^{-1} (x+h)}[/tex]
Now,
We know that,
[tex]tan^{-1}(A - B) = tan^{-1}(\frac{A - B}{1 + AB} )[/tex]
[tex]F'(x) = \lim_{h \to 0} \frac{x}{h} (tan^{-1} (x+h) - tan^{-1} x) + tan^{-1} (x+h)}\\= \lim_{h \to 0} \frac{x}{h} tan^{-1} (\frac{(x+h) - x}{1 + (x+h)x}) + tan^{-1} (x+h)}\\= \lim_{h \to 0} \frac{x}{h} tan^{-1} (\frac{h}{1 + (x+h)x}) + tan^{-1} (x+h)}\\\\= \lim_{h \to 0} \frac{x}{h\frac{1 + x(x+h)}{1 +x(x+h)} } tan^{-1} (\frac{h}{1 + (x+h)x}) + tan^{-1} (x+h)}\\= \lim_{h \to 0} \frac{x}{{1 +x(x+h)} }.\frac{1 +x(x+h)}{h} tan^{-1} (\frac{h}{1 + (x+h)x}) + tan^{-1} (x+h)}[/tex]
Now,
We know that,
[tex]\lim_{h \to 0} \frac{tan^{-1} x}{x} = 1[/tex]
∴ we get
[tex]F'(x) = \lim_{h \to 0} \frac{x}{{1 +x(x+h)} }.\frac{1 +x(x+h)}{h} tan^{-1} (\frac{h}{1 + (x+h)x}) + tan^{-1} (x+h)}\\ = \lim_{h \to 0} \frac{x}{{1 +x(x+h)} }.\lim_{h \to 0}\frac{1 +x(x+h)}{h} tan^{-1} (\frac{h}{1 + (x+h)x}) + \lim_{h \to 0}tan^{-1} (x+h)}\\= \frac{x}{{1 +x(x+0)} }.1 + tan^{-1} (x+0)}\\= \frac{x}{{1 +x(x)} } + tan^{-1} (x)}\\= \frac{x}{{1 +x^{2} } } + tan^{-1} (x)}[/tex]
So,
We get
[tex]F'(x) = \frac{x}{{1 +x^{2} } } + tan^{-1} (x)}[/tex]
problematic substances mechanical engeneering???
Answer:
The climate crisis. ...
Making water clean and accessible. ...
Providing enough food. ...
Personalised and relevant education. ...
Improving health care. ...
The refugee crisis. ...
Cyber security. ...
Enlisting the youth
Answer:
providing enough food for the African kids and families
Explanation:
Which of the following is typical of the selection of survey participants? They can indicate partiality toward a group. They must be surveyed more than once. They can be selected on purpose by a researcher. They must be representative of the population.
Answer:
Option D
Explanation:
In general a sample collected from a set of naturally distributed data set must also follow the bell curve or natural distribution
In other way round it can be said that sample set must have the same characteristics as that of the main set and hence a sample must be representative of the population.
Option D is correct
Answer:
the answer is they must be representative of the population.
Explanation:
I took the test and got it right.
Who else goes to Loveland High school?
Answer:
i dont go there lol and omg i wouldent be telling random people were you go to school it aint safe
Explanation:
Answer:one of my friends
Explanation:
my friend leo goes to it im soon gonna go to it.
Why are the more strict tools for assessing green products those that demand second-or third-party verification?
O The assessment process is more adaptable.
O The assessment process takes longer.
The assessment process iphore independent.
The assessment process is more elaborate.
Answer:
C,
The assessment process is more independent.
Explanation:
PLS ANSWER QUICKLY
Communication can take the form of: A.Spoken/verbal, non-verbal, electronic, written B.Spoken/verbal, non-verbal, written, visuals C.Effective, verbal, non-verbal, electronic D.Effective, ineffective, verbal, written
Answer:
Option B
Explanation:
Communication can be of following types -
a) Verbal - Through Speaking
b) Non Verbal - Through expression
c) Writing - in written form
d) Visuals - by seeing
Hence, option B is the right choice of answer
miles (322,000 kilometers [km]) or
Typically, if well maintained, an automotive engine will last many years and
more.
Select one:
a. 100,000
b. 150,000
c. 200,000
d. 250,000
Match the thermal conductivity to the material by drawing a line between them.
Glass Aluminum Insulating Material
205 W/m.K 0.035 W/m.K 0.935 W/m.K
Answer:
Glass → 0.935 W/(m·K)
Aluminum → 205 W/(m·K)
Insulating Material → 0.035 W/(m·K)
Explanation:
Thermal conduction is the motion of molecules of a body, randomly, across a segment or region that has a temperature gradient that results in the transportation of energy
The thermal conductivity of a material is a measure of the ability of the material to transfer or conduct heat energy
Due to the availability of moving charged particles in metals, metals conduct heat more than non-metals as we can use metal pots for cooking but we cannot readily cook using non metallic material using thermal materials directly on a stove
Therefore, aluminum has a higher thermal conduction than glass and glass will have a higher thermal conduction than an insulating material
We therefore have;
Aluminum → 205 W/(m·K)
Glass → 0.935 W/(m·K)
Insulating Material → 0.035 W/(m.K).