Question

find nth derivative of a^x.cosx
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Answer 1

please refer attachment

Answer 2

Answer:

This is the nth derivative of a^x.cosx

Step-by-step explanation:

As per the data given in the question

we have to calculate

nth derivative of a^x.cosx

As per the question

It is given that a^x.cosx.

By using complex number we have to find nth derivative.

write a^x.cosx = $a^{x}ka(a^{ix})=ka(a^{(1+i)x} )$

calculate nth derivative of $a^{(1+i)x}$

$\frac{da^{(1+i)x} }{dx}= (1+i)^{n}a^{(1+i)x}$

Taking real part

$\frac{d a^{x} cos x }{dx}=ka(1+i)^{n}a^{(1+i)x}$

To simplify that we have to write

1+i = $\sqrt{2}a^{\frac{i\pi }{4} }$  trigonometric form of 1+i

$(1+i)^n a^{ix}=\sqrt{2}a^{i(\frac{n\pi }{4}+x) }$

Finally taking real part

$\frac{d a^{x} cos x }{dx}=2^{\frac{n}{2} }a^{x}cos (\frac{n\pi }{4}+x)$

Hence, this is the nth derivative of a^x.cosx.

For such type of question:

https://brainly.in/question/54336561

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