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

Unlocking the Secrets of Curves: Higher Order Derivatives in Graph Analysis: this Effortless Math guide explains the topic in plain language, shows how it works through solved examples, and gives you free practice to try immediately, so the idea sticks instead of staying abstract.

Worked example: repeated derivatives of √(3x + 5)

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.

What the graphs show

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:

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