[qdeck ” bold_text=”false”]

[h]SL 5.7 The second derivative

[q] **Second Derivatives**

As you differentiate a function to find its derivative, you can also differentiate a derivative, and that gives you the second derivative \([f”(x)\] or \(\frac{d^2y}{dx^2}\). This has meaning too: If derivatives can tell you the slope/gradient/rate of change of a curve at a point, then the second derivative tells you the ‘rate of change of the gradient’.

Example 1: If \(f(x) = 6x^3 – 10x^2 + 8\), find \(f”(x)\):

We find the first derivative: \(f'(x) = 18x^2 – 20x\). Then, differentiate again:

\(f”(x) = 36x – 20\).

[q]

GRAPHS OF \(f'(x)\) & \(f”(x)\):

If the derivatives are just new functions, then they can be graphed as well. We know various things about the relationship between a function and its derivative. Such as:

– An increasing region of \(f(x)\) is a positive part of \(f'(x)\), or that a turning point, in \(f(x)\) is where \(f’ = 0\), i.e., an x-intercept of \(f'(x)\).

Example: Take the example below. So, some of those facts are seen in the three graphs below:

– \(x_1\): In \(f(x)\), this is a maximum point. In \(f'(x)\), it equals 0, it is a root. In \(f”(x)\), it is negative at \(x_1\).

– \(x_2\): In \(f(x)\), this is an inflection point, a min. point of \(f(x)\), & a root of \(f”(x)\) [see all of this in 5.8].

– \(x_3\): In \(f(x)\), this is a minimum point. In \(f'(x)\), it equals 0, it is a root. In \(f”(x)\), it is positive at \(x_3\).

[x] Exit text

(enter text or “Add Media”; select text to format)

[/qdeck]