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How do you differentiate #f(x)= (x) / (csc(x)+8)#?

Answer 1

Christopher Allers

f(x)=#((cscx+8)-x(-cscx*cotx +0))/(csc+8)^2#

using quotient rule: #d((f(x)/g(x)))/dx=(g(x)*d(f(x))/dx - f(x)*d(g(x))/dx)/(g(x))^2 #

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Answer 2

Adam Adkins

To differentiate the function ( f(x) = \frac{x}{\csc(x) + 8} ), you can use the quotient rule, which states that if you have a function in the form ( \frac{u(x)}{v(x)} ), the derivative is given by ( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ). Applying this rule, the derivative of ( f(x) ) with respect to ( x ) is:

[ f'(x) = \frac{(1)(\csc(x) + 8) - (x)(-\csc(x)\cot(x))}{(\csc(x) + 8)^2} ]

Simplifying the expression yields:

[ f'(x) = \frac{\csc(x) + 8 + x\csc(x)\cot(x)}{(\csc(x) + 8)^2} ]

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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.