Question Video: Finding the Value of the Derivative of a Function at a Point given the Equation of the Tangent to the Curve at That Point Mathematics

Video Transcript

The line 𝑦 equals five 𝑥 plus four is tangent to the graph of function 𝑓 at the point negative one, negative one. What is 𝑓 prime of negative one?

We’re asked to evaluate the derivative of this function 𝑓 at the point where its 𝑥-coordinate is equal to negative one. But we don’t know anything about the function, apart from the fact that the line 𝑦 equals five 𝑥 plus four is tangent to its graph at this point. We don’t know the shape of the graph of our function 𝑓. We could assume arbitrarily that it is perhaps a cubic graph. But because we do know that it has this tangent 𝑦 equals five 𝑥 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 𝑦 equals 𝑓 of 𝑥 at the point negative one, negative one will be the same as the slope of its tangent at that point. That’s the line 𝑦 equals five 𝑥 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 𝑦 equals 𝑚𝑥 plus 𝑏, then the value of 𝑚, the coefficient of 𝑥, gives the slope of the line and the value of 𝑏, the constant term, gives its 𝑦-intercept. Our tangent has been given in this form. It’s 𝑦 equals five 𝑥 plus four. So by comparing this with the general form, we can see that the slope of the line 𝑦 equals five 𝑥 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 we’re asked to find 𝑓 prime of negative one, this means we’re being asked to find the slope of the curve at the point where the 𝑥-coordinate is equal to negative one. And we’ve just seen that the slope of the tangent to the curve at this point — and hence the slope of the curve itself — is five.

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