### Video Transcript

Find the slope of the tangent line to π equals π at π equals π by two.

Weβve been given an equation to a polar curve, π is equal to some function of π. And weβre asked to find the slope of the tangent line to the curve at a given point, where π is equal to π by two. To find the slope, we need to find the derivative of π¦ with respect to π₯. And so, we use a special formula. This formula says that dπ¦ by dπ₯ is equal to dπ by dπ sin π plus π cos π all over dπ by dπ cos π minus π sin π. We can see weβre going to need to begin then by working out dπ by dπ. π is equal to π, and the derivative of π with respect to π is simply one. So we say that dπ by dπ is equal to one. Replacing dπ by dπ in our equation for the slope, we get one sin π plus π cos π over one cos π minus π sin π. And, of course, we donβt really need these ones.

But weβre also told that π is equal to π. So letβs replace π with π and we have an equation for dπ¦ by dπ₯ purely in terms of one variable, in terms of π. The slope of the tangent line to π equals π is given by the value of dπ¦ by dπ₯ at the point where π is equal to π by two. So we substitute π equals π by two into sin π plus π cos π over cos π minus π sin π. Now, of course, sin of π by two is one. Cos of π by two is zero. And we repeat these values on the denominator of our fraction. So we get one plus zero as our numerator. And our denominator is zero minus π by two, which simplifies to one over or divided by negative π by two.

Now, of course, to divide by a fraction, we multiply by the reciprocal of that fraction. So dπ¦ by dπ₯ evaluated at π equals π by two is one multiplied by negative two over π, which is simply negative two over π. And so, the slope of the tangent line to π equals π at π equals π by two is negative two over π.