How do you differentiate #y=csctheta(theta+cottheta)#?
we will need to use the product rule
tiding up.
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To differentiate ( y = csc(\theta) (\theta + cot(\theta)) ), we can use the product rule and the derivatives of the cosecant and cotangent functions:
Product Rule: ( (uv)' = u'v + uv' )
Derivative of cosecant: ( \frac{d}{d\theta} csc(\theta) = -csc(\theta) cot(\theta) )
Derivative of cotangent: ( \frac{d}{d\theta} cot(\theta) = -csc^2(\theta) )
Now, let's differentiate ( y ):
[ y' = \frac{d}{d\theta} \left[csc(\theta) (\theta + cot(\theta))\right] ]
[ = \frac{d}{d\theta} [csc(\theta)] (\theta + cot(\theta)) + csc(\theta) \frac{d}{d\theta} (\theta + cot(\theta)) ]
[ = (-csc(\theta) cot(\theta))(\theta + cot(\theta)) + csc(\theta)(1 - csc^2(\theta)) ]
[ = -csc(\theta) \cdot cot(\theta) \cdot \theta - csc(\theta) \cdot cot^2(\theta) + csc(\theta) - csc^3(\theta) ]
[ = -\theta csc(\theta) cot(\theta) - csc(\theta) \cdot \frac{cos^2(\theta)}{sin^2(\theta)} + csc(\theta) - \frac{1}{sin^3(\theta)} ]
[ = -\theta csc(\theta) cot(\theta) - \frac{cos^2(\theta)}{sin(\theta)} + csc(\theta) - \frac{1}{sin^3(\theta)} ]
[ = -\theta csc(\theta) cot(\theta) - \frac{1 - sin^2(\theta)}{sin(\theta)} + csc(\theta) - \frac{1}{sin^3(\theta)} ]
[ = -\theta csc(\theta) cot(\theta) - \frac{1}{sin(\theta)} + \frac{sin(\theta)}{sin(\theta)} + csc(\theta) - \frac{1}{sin^3(\theta)} ]
[ = -\theta csc(\theta) cot(\theta) - \frac{1}{sin(\theta)} + \frac{1}{sin(\theta)} + csc(\theta) - \frac{1}{sin^3(\theta)} ]
[ = -\theta csc(\theta) cot(\theta) + csc(\theta) - \frac{1}{sin^3(\theta)} ]
So, the derivative of ( y ) with respect to ( \theta ) is:
[ y' = -\theta csc(\theta) cot(\theta) + csc(\theta) - \frac{1}{sin^3(\theta)} ]
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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.
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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