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Allushta [10]
3 years ago
9

Find the derivative of sinx/1+cosx, using quotient rule​

Mathematics
1 answer:
Mrrafil [7]3 years ago
4 0

Answer:

f'(x) = -1/(1 - Cos(x))

Step-by-step explanation:

The quotient rule for derivation is:

For f(x) = h(x)/k(x)

f'(x) = \frac{h'(x)*k(x) - k'(x)*h(x)}{k^2(x)}

In this case, the function is:

f(x) = Sin(x)/(1 + Cos(x))

Then we have:

h(x) = Sin(x)

h'(x) = Cos(x)

And for the denominator:

k(x) = 1 - Cos(x)

k'(x) = -( -Sin(x)) = Sin(x)

Replacing these in the rule, we get:

f'(x) = \frac{Cos(x)*(1 - Cos(x)) - Sin(x)*Sin(x)}{(1 - Cos(x))^2}

Now we can simplify that:

f'(x) = \frac{Cos(x)*(1 - Cos(x)) - Sin(x)*Sin(x)}{(1 - Cos(x))^2} = \frac{Cos(x) - Cos^2(x) - Sin^2(x)}{(1 - Cos(x))^2}

And we know that:

cos^2(x) + sin^2(x) = 1

then:

f'(x) = \frac{Cos(x)- 1}{(1 - Cos(x))^2} = - \frac{(1 - Cos(x))}{(1 - Cos(x))^2} = \frac{-1}{1 - Cos(x)}

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3 years ago
Decide whether each table could represent a proportional relationship. If the
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Answer:

Step-by-step explanation:

If given tables in the picture show the proportional relationship,

Number of wheels (w) ∝ Number of buses (b)

w ∝ b

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Number of buses (b)           Number wheels (w)             Wheels per bus (\frac{w}{b})

          5                                           30                               \frac{30}{5}=6              

          8                                           48                               \frac{48}{8}=6

         10                                           60                               \frac{60}{10}=6

         15                                           90                               \frac{90}{15}=6

Here, proportionality constant is 6.

Similarly, If number of wheels (w) ∝ Number of train cars (t)

w = kt

Here, k = proportionality constant

k = \frac{w}{t}

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              20                                        184                           \frac{184}{20}=9.2

              30                                        264                           \frac{264}{30}=8.8

              40                                        344                           \frac{344}{40}=8.6

              50                                        424                           \frac{424}{50}=8.48

Since, ratio of w and t is not constant, relation between number of wheels and number of train cars is not proportional.

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