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

Answer 1

Aurora Alvarado

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:

#d/(dx) u(x)v(x) = u'(x)v(x)+u(x)v'(x)#

So with #u(x) = (sin x + x)# and #v(x) = (x + e^x)#, we find:

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

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Answer 2

Ezekiel Alexander

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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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.