1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
ad-work [718]
3 years ago
15

Explain the derivation behind the derivative of sin(x) i.e. prove f'(sin(x)) = cos(x)

Mathematics
2 answers:
ziro4ka [17]3 years ago
7 0
1.

f'(\sin x) =  \lim_{h \to 0}  \frac{f(x+h) - f(x)}{h}  =    \lim_{h \to 0}  \frac{\sin(x+h) - \sin(x)}{h}  =  \\  \\  =   \lim_{h \to 0}  \frac{2 \sin( \frac{x+h - x}{2}) \cdot \cos( \frac{x+h+x}{2})  }{h} =   \lim_{h \to 0}    \frac{2 \sin( \frac{h}{2}) \cos( \frac{2x+h}{2} ) }{h}   =  \\  \\   = \lim_{h \to 0}     [ \frac{\sin( \frac{h}{2}) }{ \frac{h}{2} }  \cdot  \cos (\frac{2x+h}{2}) ] =   \lim_{h \to 0} [1 \cdot \cos( \frac{2x+h}{2} )  ] =

= \cos( \frac{2x}{2}) = \boxed{\cos x}

2.

f'(\cos x) =  \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} =   \lim_{h \to 0}  \frac{\cos(x+h) - \cos(x)}{h}  =  \\  \\  =   \lim_{h \to 0}  \frac{-2 \sin ( \frac{x+h+x}{2}) \cdot \sin ( \frac{x+h-x}{2})  }{h}  =   \lim_{h \to 0}  \frac{-2 \sin ( \frac{2x+h}{2}) \cdot \sin ( \frac{h}{2})  }{h}  =  \\  \\  =     \lim_{h \to 0}   \frac{-2 \sin ( \frac{2x+h}{2}) }{2}     \cdot  \frac{sin( \frac{h}{2}) }{ \frac{h}{2} }    =   \lim_{h \to 0}  -\sin( \frac{2x+h}{2}) \cdot 1 =

= -\sin(  \frac{2x}{2}) = \boxed{\sin x }

3.

f'(\tan) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\tan(x+h) - \tan(x)}{h} = \\ \\ = \lim_{h \to 0} \frac{ \frac{\sin(x+h-x)}{\cos(x+h) \cdot \cos(x)} }{h} = \lim_{h \to 0} \frac{ \frac{\sin(h)}{ \frac{\cos(x+h-x) + \cos(x+h+x)}{2} } }{h} =

= \lim_{h \to 0} \frac{ \frac{\sin(h)}{\cos(h) + \cos(2x+h)} }{ \frac{1}{2}h } = \lim_{h \to 0} \frac{\sin(h)}{ \frac{1}{2}h \cdot [\cos(h) + \cos(2x+h)] } = \\ \\ = \lim_{h \to 0} \frac{\sin(h)}{h} \cdot \frac{1}{ \frac{1}{2} \cdot (\cos(h) + cos(2x+h) } = 1 \cdot \frac{1}{ \frac{1}{2} \cdot (1+ cos(2x) } = \frac{2}{1 + 2 \cos^{2} - 1 } = \\ \\ = \frac{2}{2 \cos^{2} x} = \boxed{ \frac{1}{\cos^{2}x} }

4.

f'(\cot) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\cot(x+h) - \cot(x)}{h} = \\ \\ = \lim_{h \to 0} \frac{ \frac{\sin(x - x - h)}{\sin (x+h) \cdot \sin (h)} }{h} = \lim_{h \to 0} \frac{ \frac{\sin(-h) }{ \frac{\cos(x+h-x) - \cos(x+h+x)}{2} } }{h} =

= \lim_{h \to 0} \frac{ \frac{-\sin(h)}{\cos(h) - \cos(2x+h)} }{ \frac{1}{2}h } = \lim_{h \to 0} \frac{ - \sin(h)}{ \frac{1}{2}h \cdot [\cos(h) - \cos(2x+h)] } = \\ \\ = \lim_{h \to 0} \frac{- \sin (h)}{h} \cdot   \frac{1}{ \frac{1}{2} \cdot [\cos(h) - \cos(2x+h)] }  = -1 \cdot  \frac{2}{1 - cos(2x)}  =  \\  \\  = - \frac{2}{1 -1 + 2 \sin^{2}x}  = - \frac{2}{2 \sin^{2} x} = \boxed{- \frac{1}{\sin^{2} x} }
Sever21 [200]3 years ago
6 0
I posted an image instead.

You might be interested in
Please i need help:/
guajiro [1.7K]

Answer:

the answer would be B or 358.38

8 0
2 years ago
State whether the following is talking about area or volume.
irga5000 [103]
1.volume*filling a shape*
2.volume
3.area
4.volume
5.area
6 0
3 years ago
What is the highest common factor of 264627 and 305613
Gemiola [76]

Answer:

The Greatest Common Factor (GCF) is:   3 x 3 x 3 x 3 x 11 = 891

6 0
3 years ago
Read 2 more answers
The product of two consecutive integers is 72.
Elis [28]

Answer:

First I would find the square root of 72 with a calculator, which happens to be 8.49 (rounded to the nearest hundreth)

So we know that the two integers will be close to 8.48

Since they are consecutive, lets first test out 8 and 9

We multiply these two and get 72, which is the number we were looking for.

now we find their sum.

9 + 8 = 17

Now since this is not part of the answers, we now asume that both 9 and 8 are negative numbers, since negative times a negative is a positive, but a negative plus a negative can still be negative.

-9 + -8 = -9 - 8 = -17!

So, the answer you are looking for is E: -17

5 0
3 years ago
Point F(2, –4) is translated using the rule (x – 3, y + 2).<br><br> What is the x-coordinate of F′ ?
9966 [12]
Subtracting 3 from 2 gives you the x-coordinate of F'   ... it is -1.
6 0
3 years ago
Other questions:
  • anyone help if you guys help me the next question i will give u more points the screenshot is attached below
    12·1 answer
  • A fire truck has a long ladder that is made of 3 sections. Each section is 43/4 meters long.
    15·1 answer
  • Rearrange the quadration equation so that it is equal to 0. Then factor the equation.
    6·1 answer
  • 25 points!
    13·2 answers
  • A delivery truck has 40,000 miles on its odometer and travels an average of 22,000 miles per year.
    10·1 answer
  • (4x + 2x + 2x2- 8) + (2x2 + x2+ 9)<br> Subtract or add the polynomials.
    8·1 answer
  • Suppose you want to study the relationship between smoking and cancer. You assume that smoking is a cause of cancer. Studies hav
    9·1 answer
  • How to Estimate √40 to the nearest tenth.
    11·1 answer
  • Give the prime algebraic factorization: 15a2b = _____
    12·1 answer
  • Nikos, a 39-year old male, bought a 300,000, 20-year life insurance policy.
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!