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To find the derivative of the function \( f(u) = u \cdot \sqrt[3]{u} \) at \( a = 1 \) using the definition of a derivative, we proceed as follows:

### Step 1: Write the definition of the derivative
The derivative of \( f(u) \) at \( u = a \) is given by:

\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
\]

Here, \( a = 1 \), so the derivative at \( u = 1 \) is:

\[
f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h}
\]

### Step 2: Calculate \( f(1) \) and \( f(1+h) \)

First, evaluate \( f(1) \):

\[
f(1) = 1 \cdot \sqrt[3]{1} = 1
\]

Next, calculate \( f(1+h) \):

\[
f(1+h) = (1+h) \cdot \sqrt[3]{1+h}
\]

### Step 3: Substitute into the derivative formula

Now substitute these into the formula for the derivative:

\[
f'(1) = \lim_{h \to 0} \frac{(1+h) \cdot \sqrt[3]{1+h} - 1}{h}
\]

### Step 4: Simplify the expression

To evaluate the limit, it's necessary to simplify the expression:

\[
f'(1) = \lim_{h \to 0} \frac{(1+h) \cdot \sqrt[3]{1+h} - 1}{h}
\]

Expand and simplify:

\[
= \lim_{h \to 0} \frac{(1+h)\left(\sqrt[3]{1+h}\right) - 1}{h}
\]

### Step 5: Evaluate the limit

To evaluate this limit rigorously, you might use a Taylor expansion or small h approximation around \( h = 0 \):

For small \( h \), \( \sqrt[3]{1+h} \approx 1 + \frac{h}{3} \). Therefore:

\[
(1+h)\cdot\left(1 + \frac{h}{3}\right) = 1 + h + \frac{h}{3} + \frac{h^2}{3} \approx 1 + \frac{4h}{3}
\]

Thus,

\[
f'(1) \approx \lim_{h \to 0} \frac{1 + \frac{4h}{3} - 1}{h} = \lim_{h \to 0} \frac{\frac{4h}{3}}{h} = \frac{4}{3}
\]

### Final Answer

So the derivative of \( f(u) = u \sqrt[3]{u} \) at \( u = 1 \) is:

\[
f'(1) = \frac{4}{3}
\]

Would you like more details on any step, or do you have any questions?

Here are five related questions for further exploration:

1. How would you find the derivative of \( f(u) = u \cdot \sqrt[3]{u} \) at any arbitrary point \( u = a \)?
2. What happens if you apply the chain rule to differentiate \( f(u) = u \cdot \sqrt[3]{u} \) directly?
3. Can you extend this method to find the second derivative of \( f(u) \) at \( u = 1 \)?
4. How does the derivative change if the function is instead \( f(u) = u \cdot \sqrt[4]{u} \)?
5. What is the geometrical interpretation of the derivative \( f'(1) = \frac{4}{3} \)?

**Tip:** Understanding how to use the definition of the derivative can help solidify your grasp of foundational calculus concepts, especially for complex functions!

Learn how to find the derivative of the function f(u) = u * cube root(u) at u = 1 using the definition of a derivative. Follow a step-by-step explanation, calculate f(1) and f(1+h), and simplify the expression to find f'(1) = 4/3. Explore related questions on derivatives and calculus concepts.

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