# How do you differentiate #f(x)=(e^x+sinx)(cot^2x+1)# using the product rule?

is the function obtained after differentiating using the product rule for the function described by

Given:

Let

In accordance with the product rule,

Here,

Differing with respect to x

According to the sum rule

Differing with respect to x

According to the sum rule

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To differentiate the function ( f(x) = (e^x + \sin(x))( \cot^2(x) + 1) ) using the product rule, follow these steps:

- Let ( u(x) = e^x + \sin(x) ) and ( v(x) = \cot^2(x) + 1 ).
- Find the derivatives of ( u(x) ) and ( v(x) ), denoted as ( u'(x) ) and ( v'(x) ), respectively.
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Substitute the expressions for ( u'(x) ), ( v'(x) ), ( u(x) ), and ( v(x) ) into the product rule formula.
- Simplify the expression to obtain the derivative ( f'(x) ).

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