By computing the first few derivatives and looking for a patte...

Question

By computing the first few derivatives and looking for a pattern, find the following derivatives.

a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)

Accepted Answer

Hello. In this video, we are going to be determining the 500th derivative of 7 cosine 3X by examining the pattern of the derivatives. Let's go ahead and start this problem by taking the 1st 4 derivatives of 7 cosine of 3 X. By the chain rule, the first derivative of 7 cosine 3X is going to be -7, multiplied by 3, sin of 3 X. By the chain rule again, the second derivative Y can be written as -7 multiplied by 32, multiplied by cosine of X. Cosine 3X. The third derivative, Y can be written as 7 multiplied by 3 cubed, multiplied by sin of 3 X, and the fourth derivative Y fourth power is going to be 7 multiplied -7 multiplied by 34th power multiplied by cosine of 3 X. And apologies derivative sine is positive cosine, so we wouldn't have a negative coefficient. OK, so what are the patterns that we notice in the 1st 4 derivatives? Well, notice that for every 4 derivatives that we take, we have this cyclic pattern. First, for the original function, we have cosine of 3 X. Then when we take the first derivative, we get negative sine of 3 X. When we take the second derivative, we get negative cosine of 3 X. Taking the third derivative gives us positive sign of 3 X, and taking the 4th derivative brings us back to the original cosine 3X. So, what this means is that every 4th derivative, the function cycles back to the original function. But notice that for every time we take the derivative, we get a multiple of 3 that is raised to the number derivative we are taking. So, for example, when we took the 2nd derivative, 3 is raised to the second power. When we take the 4th derivative, 3 is raised to the 4th power. And that means that each iteration is going to be multiplied by 3 to the end, where N is the number of cycles. Let's go ahead and divide the number 500 by 4. 500 divided by 4 is going to be 125 with the remainder of 0. What this means is that when we cycle around 500 times, the function is going to be back to its original cosine of 3 X. Furthermore, this 500th derivative is going to have a power of 3 raised to the 500th power. And since every multiple or every derivative has a multiple of 7, with the original one being positive, that means the 500th derivative Y 500 is going to equal to 7 multiplied by 3 to 500th power multiplied by cosine of 3 X. And this is going to be the solution to the problem. So I hope this video helps in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

By computing the first few derivatives and looking for a pattern, find the following derivatives.

a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)

Key Concepts

Derivatives of Trigonometric Functions

Higher-Order Derivatives

Pattern Recognition in Derivatives

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