Question

Find the given derivative by finding the first few derivatives and observing the pattern that occurs. d 119/dx 119(sin x)

Answer 1

The 119th derivative of 119(sin x) follows a pattern based on the derivatives of the sine function. It will alternate between sine and cosine functions with respective signs. The 119th derivative is thus -119 cos(x).

Step-by-step explanation:

We are asked to find the derivative of the function 119(sin x) concerning x. To do this, we observe a pattern by finding the first few derivatives of the sine function. The derivative of sin x is cos x, and the derivative of cos x is -sin x, and so on, alternating between cos x and sin x with alternating signs.

Therefore, if we take the derivative n times, the pattern that emerges can be described as follows:

• The 1st derivative of 119(sin x) is 119 cos(x)
• The 2nd derivative of 119(sin x) is -119 sin(x)
• The 3rd derivative of 119(sin x) is -119 cos(x)
• The 4th derivative of 119(sin x) is 119 sin(x)

The pattern is that every 4th derivative will be the same as the original function, and since the 119th derivative is a multiple of 4 plus 3, the 119th derivative will be the same as the 3rd derivative of the function, which is -119 cos(x).