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

Apply the product rule to find:

#d/(dx) f(x) = (cos x + 1)(x+e^x) + (sin x + x)(1 + e^x)#

As per the product rule, we can infer:

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To differentiate ( f(x) = (sinx + x)(x + e^x) ) using the product rule, follow these steps:

- Identify the functions ( u(x) ) and ( v(x) ) in ( f(x) = u(x) \cdot v(x) ).
- Differentiate ( u(x) ) with respect to ( x ) to find ( u'(x) ).
- Differentiate ( v(x) ) with respect to ( x ) to find ( v'(x) ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Substitute the derivatives and the original functions into the product rule formula.
- Simplify the expression to get the final answer.

Using the product rule:

[ f'(x) = (\sin(x) + x)(1 + e^x) + (\cos(x) + 1)(x + e^x) ]

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