Find the indicated derivative. d 52 y / d x 52 d^{52}y/dx^{52} of y = cos x y=\cos x

Here, we're asked to find the 52nd derivative of cosine. Now this, of course, does not mean that we need to take the derivative 52 times because we know that every single fourth derivative of cosine, 4, 8, 12, all of those derivatives that are multiples of 4, are just going to be equal to cosine again. So we just need to take that 52 divided by 4 and then take derivatives based on our remainder. Now 52 divided by 4 is going to give us a perfect 13 with no remainder because 4 times 13 is 52. So what happens when our remainder is 0? Well, that means that we don't need to take any derivatives here. And our 52nd derivative, d52dx to the power of 52 of cosinex, x. That's what we're asked to find here is simply just equal to that same function, cosine x, no derivatives needed because 52 is a perfect multiple of 4. Now let me know if you have any questions here, and I'll see you in the next video.

Find the indicated derivative. d 52 y / d x 52 d^{52}y/dx^{52} of y = cos ⁡ x y=\cos x

− cos ⁡ x -\cos x − cos x

− sin ⁡ x -\sin x − sin 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.
$d^{52} y / d x^{52}$ of $y = \cos x$

$\cos x$

$\sin x$

$-\cos x$

$-\sin x$

Verified step by step guidance

Identify the function for which you need to find the 52nd derivative. Here, the function is $ y = \cos x $.

Recall that the derivatives of $ \cos x $ follow a cyclical pattern: $ \frac{d}{dx}(\cos x) = -\sin x $, $ \frac{d^2}{dx^2}(\cos x) = -\cos x $, $ \frac{d^3}{dx^3}(\cos x) = \sin x $, and $ \frac{d^4}{dx^4}(\cos x) = \cos x $. This cycle repeats every four derivatives.

Determine the position of the 52nd derivative within this cycle. Since the cycle repeats every 4 derivatives, calculate $ 52 \mod 4 $ to find the equivalent position in the cycle.

Calculate $ 52 \mod 4 $, which equals 0. This means the 52nd derivative corresponds to the same result as the 4th derivative in the cycle.

Conclude that the 52nd derivative of $ y = \cos x $ is the same as the 4th derivative, which is $ \cos x $.