Video Transcript
Find the equation of the tangent to the curve 𝑦 equals negative two 𝑥 cubed plus eight 𝑥 squared minus 19 at 𝑥 equals two.
We’re looking for the equation of the tangent to this cubic curve. And we know that the slope of the tangent to the curve 𝑦 equals 𝑓 of 𝑥 at 𝑥 equals 𝑎 is the derivative 𝑓 prime evaluated at 𝑎. So the slope of the tangent whose equation we want to find is the value of 𝑑𝑦 by 𝑑𝑥 at 𝑥 equals two.
Hopefully, having found the slope of the tangent, we’ll be well on our way to finding the equation of the tangent. This suggests that we should find 𝑑𝑦 by 𝑑𝑥, which is the derivative with respect to 𝑥 of negative two 𝑥 cubed plus eight 𝑥 squared minus 19. We can use the fact that the derivative of a sum or difference of functions is the sum or difference as appropriate of the derivatives of the functions to split this derivative into three. And we can evaluate these derivatives one by one starting with the derivative with respect to 𝑥 of negative two 𝑥 cubed.
To find this derivative, we use the fact that the derivative with respect to 𝑥 of a power of 𝑥, 𝑥 to the 𝑛, is 𝑛 times 𝑥 to the 𝑛 minus one. And as the derivative of a number times a function is that number times the derivative of the function, the derivative with respect to 𝑥 of 𝑎 times 𝑥 to the 𝑛 is 𝑎 times 𝑛𝑥 to the 𝑛 minus one. So the derivative of negative two times 𝑥 to the three is negative two times three times 𝑥 to the three minus one, which is negative six 𝑥 squared.
We can use the same rule to find the derivative of eight 𝑥 squared or eight 𝑥 to the two. This is 16𝑥. And the derivative of the constant function 19 with respect to 𝑥 is just zero. This constant term doesn’t contribute to our derivative. And so 𝑑𝑦 by 𝑑𝑥 is negative six 𝑥 squared plus 16𝑥.
The slope of our tangent is 𝑑𝑦 by 𝑑𝑥 evaluated at 𝑥 equals two. So substituting two for 𝑥, we get negative six times two squared plus 16 times two, which is eight.
Now that we found the slope of the tangent, let’s clear some room and find the equation of the tangent. We want to find the equation of the tangent line. And we know that the slope of this line is eight. But we’re going to need some other information to work out what the equation is.
The tangent’s line touches or intersects the curve 𝑦 equals negative two 𝑥 cubed plus eight 𝑥 squared minus 19 when 𝑥 is two. So when 𝑥 is two, its 𝑦-coordinates must be the same as that of the curve. Here’s a quick sketch to show why this fact is true in general.
Applying this general fact to our example, we see that our tangent passes through the point two, negative two times two cubed plus eight times two squared minus 19. Here we’ve just substituted two for 𝑥. Evaluating this, we find that the tangent passes through the point two, negative three.
And now with these two pieces of information, we have enough information to find the equation of the tangent. We can use the point slope form of the equation of a line and substitute in. The slope 𝑚 is eight. And 𝑥 naught and 𝑦 naught are two and negative three, respectively, because the tangent passes through the point two, negative three.
All we have to do now is simplify. On the left-hand side, minus negative three becomes plus three. And on the right-hand side, we expand to get eight 𝑥 minus 16. Rearranging the equation so that all the terms fall on the left-hand side, we find that the equation of the tangent to the curve 𝑦 equals negative two 𝑥 cubed plus eight 𝑥 squared minus 19 at 𝑥 equals two in standard form is 𝑦 minus eight 𝑥 plus 19 equals zero.
