what is the nth derivative of the e^x
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f(x)=xe2x
f'(x)=e2x+2xe2x=e2x(2x+1)
f''(x)=2e2x(2x+1)+e2x⋅2=2e2x(2x+2)
f'''(x)=4e2x(2x+2)+2e2x⋅2=4e2x(2x+3)
f(4)(x)=8e2x(2x+3)+8e2x=8e2x(2x+4)
Looks like
f(n)(x)=2n−1e2x(2x+n)
It should be straightforward to prove by induction on n.
f(x)=xe2x
f'(x)=e2x+2xe2x=e2x(2x+1)
f''(x)=2e2x(2x+1)+e2x⋅2=2e2x(2x+2)
f'''(x)=4e2x(2x+2)+2e2x⋅2=4e2x(2x+3)
f(4)(x)=8e2x(2x+3)+8e2x=8e2x(2x+4)
Looks like
f(n)(x)=2n−1e2x(2x+n)
It should be straightforward to prove by induction on n.
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