# If #f(x)=sin2x# then what is the #75^(th)# derivative?

# f^((75))(x) = -2^75 cos2x #

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -37778931862957161709568 cos2x #

Should we possess:

# f^((n))(x) = {

(2^nsin2x, n mod 4 = 0), (2^ncos2x, n mod 4 = 1), (-2^nsin2x, n mod 4 = 2), (-2^ncos2x, n mod 4 = 3) :} #

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To find the 75th derivative of ( f(x) = \sin(2x) ), we can use the fact that the derivative of ( \sin(ax) ) with respect to ( x ) is ( a\cos(ax) ). Repeatedly applying this rule, we see that the derivatives of ( f(x) ) will alternate between ( \sin(2x) ) and ( \cos(2x) ), with a factor of ( 2^n ) for every ( n ) derivatives taken.

Since the derivative of ( \sin(2x) ) is ( 2\cos(2x) ) and the derivative of ( \cos(2x) ) is ( -2\sin(2x) ), every even derivative will be of the form ( 2^n\sin(2x) ) and every odd derivative will be of the form ( 2^n\cos(2x) ).

For the 75th derivative:

- Since 75 is odd, the 75th derivative will be of the form ( 2^{37}\cos(2x) ).

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To find the 75th derivative of ( f(x) = \sin(2x) ), we can use the fact that the derivative of ( \sin(ax) ) with respect to ( x ) is ( a\cos(ax) ).

Given that ( f(x) = \sin(2x) ), the first derivative ( f'(x) ) would be ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

Continuing to differentiate ( f(x) = \sin(2x) ) with respect to ( x ), the second derivative ( f''(x) ) would be ( \frac{d}{dx} (2\cos(2x)) = -4\sin(2x) ).

We can observe a pattern here: the derivatives of ( \sin(2x) ) alternate between ( \sin(2x) ) and ( \cos(2x) ), with the coefficient oscillating between ( 2 ) and ( -2 ) and the sine and cosine functions swapping places. This pattern repeats every second derivative.

Since the 75th derivative of ( \sin(2x) ) will be ( \sin(2x) ) if ( 75 ) is odd, and ( \cos(2x) ) if ( 75 ) is even, we need to determine the parity of ( 75 ).

( 75 ) is an odd number, so the 75th derivative of ( \sin(2x) ) will be ( \sin(2x) ).

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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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