### Video Transcript

Calculate π prime of π , and find the vector form of the equation of the tangent line πΏ at π of zero for π of π equals cos of two π , sin of two π , π .

We can think of this as a vector-valued function. And the first thing weβre asked to do is to find π prime of π . Thatβs the first derivative of this function. Now, this is fairly straightforward. All we need to do to find the derivative of a vector function is take the derivative of each of our components. Thatβs the derivative of cos of two π , the derivative of sin of two π , and the derivative of π , all with respect to π .

So, weβll begin by quoting the general result for the derivative of cos of ππ₯. Itβs negative π sin of ππ₯. And this means that the derivative of cos of two π , with respect to π , is a negative two sin of two π . Similarly, the general result for the derivative of sin ππ₯ is π cos ππ₯. So, d by dπ of sin two π is two cos of two π .

We also know that we can differentiate a polynomial term by multiplying it by its exponent and then reducing the exponent by one. Well, the exponent for π is technically one. So, itβs one times π to the power of zero. But π to the power of zero is one. So, the first derivative of π with respect to π is just one. And this means, π prime of π is equal to negative two sin two π , two cos two π , one.

Our next job is to find the vector form of the equation of the tangent line πΏ at π of zero. In other words, when our parameter π is equal to zero. Well, letβs recall the vector equation for a straight line. A line that passes through a point π which has position vector π nought and has a direction vector π£ is given by π equals π nought plus π‘ times π£. So, weβre going to need to do two things. Weβre going to need to work out a point that our tangent line passes through and its direction of travel.

Well, we can find the direction of travel by evaluating the derivative when our parameter π is equal to zero. So, thatβs π prime of zero. π prime of zero is negative two sin of two times zero, two cos of two times zero, one. Thatβs zero, two, one. We could alternatively write this as π£ equals two π plus π.

But how do we find the point that our tangent line passes through? Well, thatβs quite straightforward too. All we need to do is evaluate our function when π , our parameter, is equal to zero. Thatβs cos of two times zero, sin of two times zero, zero. Which is one, zero, zero. And so, we see that the vector form of the equation of our tangent line is one, zero, zero plus π‘ times zero, two, one.