Find the 999th derivative of cos x.
I got sin x.
1st derivative: -sinx
2nd derivative: -cosx
3rd derivative: sinx
4th derivative: cosx
5th derivative: -sinx
..
14th derivative:
etc.
notice we are in a cyclical loop of 4 results,
notice 999 leaves a remainder of 3 when divided by 4, so the result is the same as
the 3rd derivative, namely
sinx
Well, it seems like you've really gotten yourself in a derivative frenzy! Let's see if we can resolve this in a fun way.
The 999th derivative of cos x is definitely a daunting task. It's so big, in fact, that even my clown car is having trouble fitting it in!
But fear not, my friend, for there's a pattern to this madness. The derivatives of cos x cycle through four functions: cos x, -sin x, -cos x, and sin x.
Since there are 999 derivatives, we need to determine which function in the cycle we'll end up with. Dividing 999 by 4, we find a remainder of 3. So, we're left with just one thing: sin x!
That's right, after all that derivative chaos, we finally arrive at the 999th derivative of cos x, which is sin x. Now you can rest easy and let out a sigh of cosine relief!
To find the 999th derivative of cos x, we can start by observing the pattern of the derivatives of cosine.
The derivatives of cosine alternate between sine and cosine. The derivatives of sine are also cyclical, repeating every four derivatives.
To compute the 999th derivative, we can break it down into two parts:
1. Find the remainder when 999 is divided by 4: 999 ÷ 4 = 249 remainder 3.
2. Use the remainder to determine the function that represents the 999th derivative.
Since the remainder is 3, the 999th derivative will be the derivative of sine after applying the cosine function three times.
Starting with the derivative of cosine (which is -sin x), we can differentiate three more times to get the 999th derivative.
1st derivative: -sin x
2nd derivative: -cos x
3rd derivative: sin x
Therefore, the 999th derivative of cos x is sin x.
To find the nth derivative of a function, you can use the power rule for differentiation. However, in this case, finding the 999th derivative of cos x using the power rule would be very tedious and time-consuming.
Alternatively, we can use the concept of periodicity to simplify the process. The cosine function has a period of 2π, which means that cos(x + 2π) = cos(x) for all x.
Since the derivative of a function with period T is also a function with period T, we can conclude that the pattern of derivatives of cos x repeats every 2π.
To see this pattern, let's calculate the first few derivatives:
df/dx(cos x) = -sin x
d^2f/dx^2(cos x) = -cos x
d^3f/dx^3(cos x) = sin x
d^4f/dx^4(cos x) = cos x
As we can observe, after taking the fourth derivative, the pattern repeats. This implies that the fifth derivative will be the same as the first derivative (-sin x), the sixth derivative will be the same as the second derivative (-cos x), and so on.
Since the 999th derivative is congruent to 999 modulo 4, we can determine its value based on the remainder when 999 is divided by 4.
999 divided by 4 gives a quotient of 249 with a remainder of 3. Thus, the 999th derivative will have the same value as the third derivative, which is sin x.
Therefore, you are correct, the 999th derivative of cos x is sin x.