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Answer:
See below
Step-by-step explanation:
[tex]f(x)=e^2x(x^3+1) = e^2x^4+x[/tex]
Considering
[tex]f'(x)=\dfrac{d}{dx}f(x)=\dfrac{d}{dx}e^2x^4+\dfrac{d}{dx}x[/tex]
Once the derivative of a constant is 1 and [tex]e^x = \dfrac{d}{dx} e^x[/tex]
Then,
[tex]\dfrac{d}{dx}f(x)=\dfrac{d}{dx}e^2x^4+\dfrac{d}{dx}x =\boxed{ e^2 4x^3+1}[/tex]
Therefore,
[tex]f'(2) = e^24\cdot 2^3 +1 = 32e^2+1[/tex]
The value of differentiation of the given function at x = 2 is f'(2) = 30e⁴.
Important information:
Differentiate the given function with respect to x.
[tex]f'(x)=e^{2x}\dfrac{d}{dx}(x^3+1)+(x^3+1)\dfrac{d}{dx}e^{2x}[/tex]
[tex]f'(x)=e^{2x}(3x^2)+(x^3+1)(2e^{2x})[/tex]
[tex]f'(x)=e^{2x}(3x^2+2x^3+2)[/tex]
Substitute [tex]x=2[/tex] in the above function.
[tex]f'(2)=e^{2(2)}(3(2)^2+2(2)^3+2)[/tex]
[tex]f'(2)=e^{4}(12+16+2)[/tex]
[tex]f'(2)=30e^{4}[/tex]
Therefore, the required value is [tex]f'(2)=30e^{4}[/tex].
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