Understanding Third Derivatives and Matrix Multiplication in Calculus

[bobsmith76]

Homework Statement

In the following problem they are taking the third derivative, then multiplying a 3 by 3 matrix. I don't understand how they progress from the first to the second to the third deriviative in the following two questions.

These are the parts of the solution that I really don't understand

What's going on? -3 sin t is not the derivative of -3 sin t

The derivative of e^t(cos t - sin t) is (e^t)(-sin t - cos t) not 2e^t sin t
The derivative of e^t(sin t + cos t) is (e^t)(cos t - sin t) not (2e^t)(cos t)

I don't understand.

 

[bobsmith76]

I looked at some more examples and they take the derivative for all the others and I understand them. I don't however understand the derivatives of the two above.

 

[Mark44]

Homework Statement

In the following problem they are taking the third derivative, then multiplying a 3 by 3 matrix. I don't understand how they progress from the first to the second to the third deriviative in the following two questions.

These are the parts of the solution that I really don't understand

What's going on? -3 sin t is not the derivative of -3 sin t

Looks like a typo to me. The middle row in the 2nd column should be -3cos(t).

The derivative of e^t(cos t - sin t) is (e^t)(-sin t - cos t) not 2e^t sin t
The derivative of e^t(sin t + cos t) is (e^t)(cos t - sin t) not (2e^t)(cos t)

Your work here is incorrect. Both functions are products, so you need to use the product rule when you differentiate.

I don't understand.

 

[bobsmith76]

excellent. i got it now. i really appreciate your help.