- Calculate the derivative of the function r of t equals t natural log of t i plus 5e to the t j plus quantity cosine t minus sine t multiplied by k. In order to find r prime of t, which is our goal, we will take the derivative of this vector valued function component wise. So we'll take the derivative of t natural log of t using our product rule.

So that would be t times 1 over t plus 1 times natural log of t. That will be our i component. Our j component will be the antiderivative of 5 e to the t, which is 5 e to the t. That's our j component. And then the derivative of cosine t minus sine t, that will be our k component. The derivative of cosine t is minus sine t, and the derivative of minus sine t is minus cosine t.

So that makes the function, r prime of t. And actually, the i component, we can simplify, so that would be 1 plus natural log of t. So that would make our entire function parentheses 1 plus natural log of t close parentheses i plus 5 e to the t j plus parentheses negative sine t minus cosine t close parentheses, k.