How do you compute the 200th derivative of #f(x)=sin(2x)#?

Answer 1

#2^200 sin(2x)#

#f(x) = sin(2x) implies# #d/dx f(x) = 2 cos(2x) implies# #d^2/dx^2 f(x) = -2^2 sin(2x) implies# #d^3/dx^3 f(x) = -2^3 cos(2x) implies# #d^4/dx^4 f(x) = 2^4 sin(2x) implies#
This means that differentiating #f(x)# 4 times results in the same function, with a multiplying factor of 4.

Differentiating 200 times is the same as repeating the above 50 times, and so

#d^200/dx^200 f(x) = (2^4 )^50sin(2x) = 2^200 sin(2x)#
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Answer 2

To compute the 200th derivative of ( f(x) = \sin(2x) ), you can use the property that the derivative of ( \sin(ax) ) with respect to ( x ) is ( a\cos(ax) ).

Given that ( f(x) = \sin(2x) ), the first derivative is:

[ f'(x) = 2\cos(2x) ]

The second derivative is:

[ f''(x) = -4\sin(2x) ]

You'll notice a pattern emerging: every even derivative will involve ( \sin(2x) ) and every odd derivative will involve ( \cos(2x) ), with a coefficient of ( 2^k ) where ( k ) is the number of times the derivative has been taken.

Since the 200th derivative will be even, it will involve ( \sin(2x) ). The coefficient for the 200th derivative will be ( 2^{100} ) (as ( 200 ) is even).

Therefore, the 200th derivative of ( f(x) = \sin(2x) ) is:

[ f^{(200)}(x) = 2^{100}\sin(2x) ]

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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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