Find the indicated derivative.d1002/dx1002(sinx)d^{1002}/dx^{1002}(\sinx)

− sin ⁡ x -\sin x − sin x

Find the indicated derivative. d 1002 / d x 1002 ( sin x ) d^{1002}/dx^{1002}(\sinx)

Here we're asked to find the 1,000 and second derivative of the sine of x. Now remember that this does not mean we need to take the derivative of 1,002 times. We can simply take that 1,000 and 2 and then divide it by 4 in order to figure out how many derivatives we need to take based on whatever our remainder is because every 4th derivative of sine is also sine. So if I take that 1,002 and divide it by 4, that's going to give me 250 because 4 times 250 gives us a 1,000. So here, my remainder is going to be 2 because I need 2 more to get from that 1,000 to 1,002. So that tells me here I need to find just my second derivative. So our 1,000 and second derivative is the same thing as just our second derivative based on what we know about higher order derivatives of sine and cosine. So here I just need to find that second derivative. So in order to do that, I, of course, need to start with that first derivative. That's ddx of sine x. Now, ddx of sine x we know is just equal to the cosine of x. So taking one more derivative here to get that second derivative, d2dx2ofsinex, taking derivative of that first derivative, cosine. The derivative of cosine is negative sine. So negative sine of x. This is our our final answer. This represents our 1,000 and second derivative of sine. Let me know if you have any questions here. I'll see you in the next one.

Find the indicated derivative. d 1002 / d x 1002 ( sin ⁡ ⁡ x ) d^{1002}/dx^{1002}(\sin⁡x)

− sin ⁡ x -\sin x − sin x

− cos ⁡ x -\cos x − cos x

3. Techniques of Differentiation

Derivatives of Trig Functions

Calculus

3. Techniques of Differentiation

Derivatives of Trig Functions

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Multiple Choice

Find the indicated derivative.
$d^{1002} / d x^{1002} (\sin x)$

$\cos x$

$\sin x$

$-\cos x$

$-\sin x$

Verified step by step guidance

Recognize that the problem involves finding the 1002nd derivative of the function $ \sin x $.

Understand that the derivatives of $ \sin x $ and $ \cos x $ form a repeating cycle: $ \sin x \rightarrow \cos x \rightarrow -\sin x \rightarrow -\cos x \rightarrow \sin x $.

Note that this cycle repeats every 4 derivatives. Therefore, to find the 1002nd derivative, determine the remainder when 1002 is divided by 4.

Calculate the remainder: 1002 divided by 4 gives a remainder of 2. This means the 1002nd derivative corresponds to the second function in the cycle.

Identify the second function in the cycle, which is $ \cos x $. Therefore, the 1002nd derivative of $ \sin x $ is $ \cos x $.

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