How do you differentiate #y= (5x)/((tanx)(cotx))#?

Answer 1

5

tan x * cot x is simply 1, because cot x equals #1/tan x#. Thus it is y= 5x. Hence #dy/dx# =5
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Answer 2

If you don't think first (before you start) you'll use the quotient rule (with the product rule to differentiate the denominator).

Take a few seconds to think, and to ask: Can I rewrite this before I differentiate to make my life easier?

#tanx = 1/cotx# so #tanx cotx =1#
That means that #y = 5x/1=5x#
So, #y'=5#
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Answer 3
#y'=5#
#y=(5x)/((tanx)(cotx)#

As

#tanx=sinx/cosx#

And

#cotx=cosx/sinx#

So,

#y=(5x)/((sinx/cosx)(cosx/sinx))#
#y=(5x)/((cancelsinx/cancelcosx)(cancelcosx/cancelsinx))#
#y=(5x)/1#
#y=(5x)#

Differentiating both sides with respect to 'x'

#y'=5#
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Answer 4

To differentiate ( y = \frac{5x}{\tan x \cdot \cot x} ), you can use the quotient rule. The quotient rule states that if you have a function of the form ( \frac{f(x)}{g(x)} ), then its derivative is given by ( \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2} ). Applying this rule to the given function:

[ y = \frac{5x}{\tan x \cdot \cot x} ]

[ f(x) = 5x ] [ g(x) = \tan x \cdot \cot x ]

Now, find the derivatives of ( f(x) ) and ( g(x) ):

[ f'(x) = 5 ] [ g'(x) = (\tan x \cdot \cot x)' ]

Using trigonometric identities, ( \tan x \cdot \cot x = \frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} = 1 ). Therefore, the derivative of ( \tan x \cdot \cot x ) is 0.

Now, apply the quotient rule:

[ y' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2} ]

[ y' = \frac{5 \cdot \tan x \cdot \cot x - 5x \cdot 0}{(\tan x \cdot \cot x)^2} ]

[ y' = \frac{5 \cdot \tan x \cdot \cot x}{(\tan x \cdot \cot x)^2} ]

[ y' = \frac{5}{\tan x \cdot \cot x} ]

[ y' = \frac{5}{1} ]

[ y' = 5 ]

Therefore, the derivative of ( y = \frac{5x}{\tan x \cdot \cot x} ) is ( y' = 5 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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