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

5

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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?

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As

And

So,

Differentiating both sides with respect to 'x'

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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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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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