Using first principles, differentiate 𝑓(𝑥) = 3𝑥^(2/3) Show your work. [Hint: use the ‘difference of cubes.’]

To differentiate \( f(x) = 3x^{2/3} \) using first principles, we need to apply the definition of the derivative. The derivative of a function \( f(x) \) is given by:

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

### Step 1: Apply the definition of the derivative
Here, \( f(x) = 3x^{2/3} \), so the derivative is:

\[
f'(x) = \lim_{h \to 0} \frac{3(x+h)^{2/3} - 3x^{2/3}}{h}
\]

Factor out the 3 to simplify:

\[
f'(x) = 3 \lim_{h \to 0} \frac{(x+h)^{2/3} - x^{2/3}}{h}
\]

### Step 2: Using the Hint (Difference of Cubes)
We know that the expression \( a^{2/3} - b^{2/3} \) can be treated similarly to the difference of cubes formula. The difference of cubes formula is:

\[
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
\]

To apply a similar idea to fractional powers like \( x^{2/3} \), we write the difference as:

\[
(x+h)^{2/3} - x^{2/3} = \frac{\left[(x+h)^{2} - x^2\right]}{(x+h)^{1/3} + x^{1/3}}
\]

### Step 3: Expanding the Numerator
Now expand the numerator \( (x+h)^2 - x^2 \):

\[
(x+h)^2 = x^2 + 2xh + h^2
\]
\[
(x+h)^2 - x^2 = 2xh + h^2
\]

Thus, we have:

\[
(x+h)^{2/3} - x^{2/3} = \frac{2xh + h^2}{(x+h)^{1/3} + x^{1/3}}
\]

### Step 4: Substitute into the Derivative
Substitute this into the derivative expression:

\[
f'(x) = 3 \lim_{h \to 0} \frac{\frac{2xh + h^2}{(x+h)^{1/3} + x^{1/3}}}{h}
\]

This simplifies to:

\[
f'(x) = 3 \lim_{h \to 0} \frac{2xh + h^2}{h \left( (x+h)^{1/3} + x^{1/3} \right)}
\]

### Step 5: Simplify the Expression
Factor out \( h \) from the numerator:

\[
f'(x) = 3 \lim_{h \to 0} \frac{h(2x + h)}{h \left( (x+h)^{1/3} + x^{1/3} \right)}
\]

Cancel the \( h \) terms:

\[
f'(x) = 3 \lim_{h \to 0} \frac{2x + h}{(x+h)^{1/3} + x^{1/3}}
\]

As \( h \to 0 \), the term \( 2x + h \) approaches \( 2x \), and \( (x+h)^{1/3} \) approaches \( x^{1/3} \). Thus, we get:

\[
f'(x) = 3 \frac{2x}{2x^{1/3}} = 3 \cdot x^{(2/3 - 1)} = 3 \cdot x^{-1/3}
\]

### Final Answer:
\[
f'(x) = \frac{3}{x^{1/3}}
\]

### Summary:
We have used first principles and the hint involving the difference of cubes to find the derivative of \( f(x) = 3x^{2/3} \). The final result is:

\[
f'(x) = \frac{3}{x^{1/3}}
\]

---

Would you like further details or clarification? Here are some additional questions you could explore:

1. How does this method compare to the power rule for differentiation?
2. What is the geometric interpretation of \( f'(x) = \frac{3}{x^{1/3}} \)?
3. How would the derivative change if the exponent were \( \frac{5}{3} \) instead of \( \frac{2}{3} \)?
4. Can this method be applied to find the second derivative of \( f(x) = 3x^{2/3} \)?
5. How would you interpret the derivative when \( x = 0 \)?

**Tip:** The difference of cubes formula is a powerful tool in simplifying complex algebraic expressions, especially when dealing with roots or fractional powers.

Using first principles, differentiate 𝑓(𝑥) = 3𝑥^(2/3). Show your work. [Hint: use the ‘difference of cubes.’]

Learn how to differentiate f(x) = 3x^(2/3) using first principles with a step-by-step solution, incorporating algebraic manipulation and the difference of cubes formula. Suitable for advanced high school and college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation