How do you differentiate #f(x)=ln2x * cot5x# using the product rule?

Answer 1

#(1/x) cot 5x +5 ln2x sec^2 5x#

It can be done using product rule of differentiation (fg)'= f'g +fg' Accordingly, required derivative would be# (1/(2x)) (2) cot 5x + ln2x sec^2 5x (5)#

=#(1/x) cot 5x +5 ln2x sec^2 5x#

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Answer 2

Christopher Agresta

To differentiate ( f(x) = \ln(2x) \cdot \cot(5x) ) using the product rule, you would follow these steps:

Identify the functions: ( u(x) = \ln(2x) ) and ( v(x) = \cot(5x) ).
Differentiate each function: ( u'(x) = \frac{1}{x} ) and ( v'(x) = -5\csc^2(5x) ).
Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
Substitute the derivatives and original functions into the formula.
Simplify the expression if necessary.

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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.