What is the nth derivative of #f(x) = x e^(2x)#?

Answer 1

We can write the derivative of this function in a ricorsive way. Ricorsive way means that also in the definition there is the concept of derivate!

#y^((1))=1*e^(2x)+x*e^(2x)*2=e^(2x)+2xe^(2x)=e^(2x)+2y#
#y^((2))=2e^(2x)+2y^((1))#
#y^((3))=2^2e^(2x)+2y^((2))#

...

#y^((n))=2^(n-1)e^(2x)+2y^((n-1))#.

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Answer 2

Thank you for providing a fun question to work on!

#f(x) = xe^(2x)# #f'(x) = e^(2x) + 2xe^(2x) = e^(2x)(2x+1)# #f''(x) = 2e^(2x)(2x+1) +e^(2x)*2 = 2e^(2x)(2x+2)# #f'''(x) = 4e^(2x)(2x+2) + 2e^(2x)*2 = 4e^(2x)(2x+3)#
#f^((4))(x) = 8e^(2x)(2x+3) + 8e^(2x) = 8e^(2x)(2x+4)#

Looks like

#f^((n))(x) = 2^(n-1)e^(2x)(2x+n)#
It should be straightforward to prove by induction on #n#.
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Answer 3

To find the nth derivative of the function ( f(x) = x e^{2x} ), we can use the general formula for finding derivatives of a product of functions. The formula states that the nth derivative of the product ( u(x)v(x) ) can be found using the Leibniz rule.

The Leibniz rule states that the nth derivative of the product of two functions u(x) and v(x) is given by the sum of terms where each term involves the nth derivative of one function and the original function itself.

For the function ( f(x) = x e^{2x} ), the nth derivative can be found using the Leibniz rule. After finding the derivatives, we notice a pattern that arises, making it possible to generalize the nth derivative.

By applying the Leibniz rule repeatedly, we find that the nth derivative of ( f(x) = x e^{2x} ) is given by:

[ f^{(n)}(x) = (2^n)(n + 1)(x e^{2x} + e^{2x}) ]

where ( f^{(n)}(x) ) denotes the nth derivative of ( f(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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