What is the derivative of #x * e^3x + tan^-1 2x#?

Answer 1

#e^(3x)+3xe^(3x)+2/(1+4x^2)#

The derivative of the expression #x.e^(3x)+tan^-1(2x)#
Knowing that : #(u+v)'=u'+v'# (1) #(e^u)'=u'e^u# (2) #(tan^-1(u))'=(u')/(1+u^2)# (3) #(u.v)'=u'v+v'u#. (4)
Lets find the derivative of #x.e^(3x)#:
#color(blue)(x.e^(3x))'# #=x'e^(3x)+x.(e^(3x))'# applying above formula (4) #=e^(3x)+x.3.e^(3x)# applying the above formula (2) #color(blue)(=e^(3x)+3xe^(3x). name it (5))#
Now let's find the derivative of #tan^-1(2x)#
#color(blue)((tan^-1(2x)))'# applying the above formula (3) #=((2x)')/(1+(2x)^2)# #color(blue)(=2/(1+4x^2) name it (6))#
The derivative of the sum #x.e^(3x)+tan^-1(2x)# is :
#color(red)((x.e^(3x)+tan^-1(2x))')# #=(x.e^(3x))'+(tan^-1(2x))'#. applying the above formula (1) #color(red)(=e^(3x)+3xe^(3x)+2/(1+4x^2)#substituting (5) and (6)
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Answer 2

To find the derivative of the given function, (x \cdot e^{3x} + \tan^{-1}(2x)), we differentiate each term separately using the rules of differentiation.

  1. For the term (x \cdot e^{3x}): Apply the product rule: [ \frac{d}{dx}\left(x \cdot e^{3x}\right) = x \cdot \frac{d}{dx}(e^{3x}) + \frac{d}{dx}(x) \cdot e^{3x} ] [ = x \cdot (3e^{3x}) + e^{3x} ] [ = 3xe^{3x} + e^{3x} ]

  2. For the term (\tan^{-1}(2x)): Apply the chain rule: [ \frac{d}{dx}(\tan^{-1}(2x)) = \frac{1}{1 + (2x)^2} \cdot \frac{d}{dx}(2x) ] [ = \frac{2}{1 + 4x^2} ]

Therefore, the derivative of the function (x \cdot e^{3x} + \tan^{-1}(2x)) is (3xe^{3x} + e^{3x} + \frac{2}{1 + 4x^2}).

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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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