Find the indicated derivative. f 19 ( x ) f^{19}(x) of f ( x ) = cos x f(x)=\cos x

this problem, we're asked to find the 19th derivative of cosine x. Now remember that this does not mean we have to take the derivative 19 times because we know that every 4th derivative of cosine x is also cosine x. So what we need to do is take that 19 and divide it by 4 and then take our derivatives based on the remainder. Now, how many times does 4 go into 19? Well, I know that 4 times 4 is 16, but 4 times 5 is 20. So I know that it's not 5, it's equal to 4. And then I have a remainder of 3 to get back up to that 19. So since I want to take derivatives based on my remainder of 3, that tells me that I need to find here my third derivative or f triple prime of x. So I need to take that derivative 3 times. Now my original function is f of x equals cosine x. So let's start by finding our first derivative here, f prime of x. F prime of x, if I take the derivative once here, the derivative of cosine is negative sine. So this is negative sine x. Then my second derivative here, f double prime of x, taking the derivative of negative sine, that's going to give me negative cosine of x. And then finally, my third derivative here, remember that we're taking it based on that remainder. So this will be our final answer. F triple prime of x, the derivative of negative cosine is going to be negative negative sine of x, which is, of course, equal to the positive sine of x. So this tells us that the 19th derivative of the cosine of x is sine x. Let me know if you have any questions here. Thanks for watching.

Find the indicated derivative. f 19 ( x ) f^{19}(x) of f ( x ) = cos ⁡ x f(x)=\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.
$the 19th power of f of x$ of $f (x) = \cos x$

$\cos x$

$\sin x$

$-\cos x$

$-\sin x$

Verified step by step guidance

Understand that the problem requires finding the 19th derivative of the function $ f(x) = \cos x $.

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

Determine the position of the 19th derivative within this cycle. Since the cycle repeats every 4 derivatives, calculate $ 19 \mod 4 $ to find the position within the cycle.

Calculate $ 19 \mod 4 $, which equals 3. This means the 19th derivative corresponds to the third derivative in the cycle.

Identify the third derivative in the cycle, which is $ \sin x $. Therefore, $ f^{(19)}(x) = \sin x $.