How do you differentiate #f(x)=(sinx+1)(x^23e^x)# using the product rule?
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To differentiate ( f(x) = (\sin x + 1)(x^2  3e^x) ) using the product rule, follow these steps:
 Identify the functions ( u(x) ) and ( v(x) ).
 Apply the product rule formula: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
 Differentiate each function ( u(x) ) and ( v(x) ) separately.
 Substitute the derivatives and original functions into the product rule formula.
 Simplify the expression.
Here's the stepbystep process:

Let ( u(x) = \sin x + 1 ) and ( v(x) = x^2  3e^x ).

Differentiate ( u(x) ) and ( v(x) ) separately:
( u'(x) = \cos x ) (derivative of ( \sin x ))
( v'(x) = 2x  3e^x ) (derivative of ( x^2  3e^x )) 
Apply the product rule formula:
( f'(x) = (\sin x + 1)(2x  3e^x) + (\cos x)(x^2  3e^x) )

Simplify the expression:
( f'(x) = 2x\sin x  3e^x\sin x + 2x  3e^x + x^2\cos x  3e^x\cos x )
So, the derivative of ( f(x) ) is ( f'(x) = 2x\sin x  3e^x\sin x + 2x  3e^x + x^2\cos x  3e^x\cos x ).
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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