### Video Transcript

The line π¦ equals five π₯ plus four is tangent to the graph of function π at the point negative one, negative one. What is π prime of negative one?

Weβre asked to evaluate the derivative of this function π at the point where its π₯-coordinate is equal to negative one. But we donβt know anything about the function, apart from the fact that the line π¦ equals five π₯ plus four is tangent to its graph at this point. We donβt know the shape of the graph of our function π. We could assume arbitrarily that it is perhaps a cubic graph. But because we do know that it has this tangent π¦ equals five π₯ plus four at the point negative one, negative one, we have some information about its slope at that point.

We know that the slope of the graph of a function is equal to the slope of its tangent at that point. So the slope of our graph π¦ equals π of π₯ at the point negative one, negative one will be the same as the slope of its tangent at that point. Thatβs the line π¦ equals five π₯ plus four. This is useful because we know how to find the slope of a straight line from its equation.

If the equation of a line is given in the form of π¦ equals ππ₯ plus π, then the value of π, the coefficient of π₯, gives the slope of the line and the value of π, the constant term, gives its π¦-intercept. Our tangent has been given in this form. Itβs π¦ equals five π₯ plus four. So by comparing this with the general form, we can see that the slope of the line π¦ equals five π₯ plus four is five.

The final piece of information we need to recall is that the slope of a curve is given by its first derivative. So when weβre asked to find π prime of negative one, this means weβre being asked to find the slope of the curve at the point where the π₯-coordinate is equal to negative one. And weβve just seen that the slope of the tangent to the curve at this point β and hence the slope of the curve itself β is five.

So we have our answer to the problem, the slope of the tangent β and hence the slope of the curve β and the value of its first derivative when π₯ is equal to negative one is five.