Question

Find the nth derivative of ex cosx

Answer 1

Given : eˣcosx

To Find : nth derivative

Solution:

eˣcosx

$e^{ix}=\cos x+i \sin x$

Cosx = Real part

$e^x.e^{ix}$

$e^{(1+i)x}$

nth derivative

= $(1 + i)^ne^{(1+i)x}$

= $e^x.(1 + i)^ne^{ix}$

1 + i  = √2 ( 1/√2 + i/√2)

= √2(Cos(π/4 + i sinπ/4)

= $\sqrt{2} . e^{i\frac{\pi}{4}$

(1 + i )ⁿ == $(\sqrt{2} . e^{i\frac{\pi}{4})^n$  = $(\sqrt{2})^n e^{i\frac{n\pi}{4}$

= $e^x.(\sqrt{2})^n e^{i\frac{n \pi}{4}e^{ix}$

=$e^x.(\sqrt{2})^n e^{i(x+\frac{n\pi}{4})}$

= eˣ (√2)ⁿ( cos(x + nπ/4) + i sin(x + nπ/4)

Taking real part only

=  eˣ (√2)ⁿ( cos(x + nπ/4))

nth derivative of eˣcosx =  eˣ (√2)ⁿ( cos(x + nπ/4))

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