https://file.aiquickdraw.com/mathbot/file/17305198285552r148dqc.jpg

Let's solve this problem step-by-step.

Given:
\[
f(x) = \sqrt[3]{x}
\]
we are asked to use the definition of the derivative:
\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
\]
to find \( f'(a) \) when \( a \neq 0 \), and then to show that \( f'(0) \) does not exist, indicating a vertical tangent line at \( (0,0) \).

### Step 1: Finding \( f'(a) \) for \( a \neq 0 \)

1. **Substitute** \( f(x) = \sqrt[3]{x} \) into the derivative formula:
   \[
   f(a) = \sqrt[3]{a} \quad \text{and} \quad f(a + h) = \sqrt[3]{a + h}
   \]
   so
   \[
   f'(a) = \lim_{h \to 0} \frac{\sqrt[3]{a + h} - \sqrt[3]{a}}{h}.
   \]

2. **Simplify the Expression Using a Conjugate Technique**:
   To handle the cube root, let’s multiply the numerator and denominator by the conjugate factor:
   \[
   f'(a) = \lim_{h \to 0} \frac{\sqrt[3]{a+h} - \sqrt[3]{a}}{h}.
   \]



Consider the following equation: Let f(x) = ³√x. If a ≠ 0, use the above formula to find f'(a). Show that f'(0) does not exist and that f has a vertical tangent line at (0,0).

Learn how to find the derivative of the cube root function, f(x) = ³√x, using the limit definition of the derivative. This problem includes verifying the non-existence of f'(0) and examining the vertical tangent line at the origin. Suitable for college calculus students.

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