How do you differentiate #f(x)=(sinx+1)(x^2-3e^x)# using the product rule?

Answer 1

#\cos (x)(x^2-3e^x)+(2x-3e^x)(\sin (x)+1)#

#\frac{d}{dx}((\sin (x)+1)(x^2-3e^x))#
Applying product rule, #(f\cdot g)^'=f^'\cdot g+f\cdot g^'#
#f=sinx +1 ,g=x^2 - 3e^x# #=frac{d}{dx}(sin (x)+1)(x^2-3e^x)+frac{d}{dx}(x^2-3e^x)(sin (x)+1)#
we know, #\frac{d}{dx}(sin (x)+1)=cos (x)#; #\frac{d}{dx}(\sin (x))=\cos(x)#; #\frac{d}{dx}(1)=0#; #\frac{d}{dx}(x^2-3e^x)=2x-3e^x#
Finally, #=\cos(x)(x^2-3e^x)+(2x-3e^x)(\sin (x)+1)#
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Answer 2

To differentiate ( f(x) = (\sin x + 1)(x^2 - 3e^x) ) using the product rule, follow these steps:

  1. Identify the functions ( u(x) ) and ( v(x) ).
  2. Apply the product rule formula: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
  3. Differentiate each function ( u(x) ) and ( v(x) ) separately.
  4. Substitute the derivatives and original functions into the product rule formula.
  5. Simplify the expression.

Here's the step-by-step process:

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

  2. 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 ))

  3. Apply the product rule formula:

    ( f'(x) = (\sin x + 1)(2x - 3e^x) + (\cos x)(x^2 - 3e^x) )

  4. 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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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.

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.

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.

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.

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