Home > Calculus > Quotient Rule

How do you differentiate #f(x)=x/cotx# using the quotient rule?

Answer 1

Isabella Ackerman

#f'(x)=(xcsc^2x+cotx)/cot^2x#

As per the Quotient Rule:

#f'(x)=(cotxd/dx[x]-xd/dx[cotx])/cot^2x#

#f'(x)=(cotx(1)-x(-csc^2x))/cot^2x#

#f'(x)=(xcsc^2x+cotx)/cot^2x#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Emily Ammerman

To differentiate ( f(x) = x/\cot(x) ) using the quotient rule:

Identify ( u(x) = x ) and ( v(x) = \cot(x) ).
Apply the quotient rule, which states that if you have two functions ( u(x) ) and ( v(x) ), then the derivative of their quotient is given by ( (u/v)' = (u'v - uv')/v^2 ).
Differentiate ( u(x) ) and ( v(x) ) separately.
Substitute the derivatives into the quotient rule formula.
Simplify the expression.

The derivative of ( f(x) = x/\cot(x) ) using the quotient rule is:

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

Simplify the expression further if necessary.

By signing up, you agree to our Terms of Service and Privacy Policy

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.