Unlocking the Secrets of Curves: Higher Order Derivatives in Graph Analysis

Let’s consider this radical function: \( f(x) = \sqrt{3x + 5} \). We’ll find the first few derivatives of this function.

1. First Derivative: To find the first derivative of \( f(x) = \sqrt{3x + 5} \), we use the chain rule, as we have a composition of functions (the square root function and the linear function \( 3x + 5 \)):
   \( f'(x) = \frac{1}{2\sqrt{3x + 5}} \cdot 3 = \frac{3}{2\sqrt{3x + 5}} \)
2. Second Derivative: Differentiating the first derivative, we get:
   \( f”(x) = \frac{d}{dx}\left( \frac{3}{2\sqrt{3x + 5}} \right) = -\frac{9}{4}(3x + 5)^{-\frac{3}{2}} \)
   This derivative involves applying the quotient rule or further application of the chain rule.
3. Subsequent Derivatives: Here are the next derivatives:

3rd: \( \frac{81}{8\left(3x+5\right)^{\frac{5}{2}}} \)

4th: \( \frac{-1215}{16\left(3x+5\right)^{\frac{7}{2}}} \)

5th: \( \frac{25515}{32\left(3x+5\right)^{\frac{9}{2}}} \)

and so on, which can go on forever.

here are the graphs for these functions, starting from the original function, \( f(x) = \sqrt{3x + 5} \), with each graph representing the “graph of changes” of the previous function at every \( x \) value: