Math Problem Statement
Using first principles, differentiate f(x) = 3x^(2/3). Show your work. [Hint: use the ‘difference of cubes.’]
Solution
To differentiate using first principles, we use the definition of the derivative:
Step 1: Substitute into the definition
We can factor out the constant :
Step 2: Use the difference of cubes formula
We know that the difference of cubes can be written as:
To use this hint, we need to rewrite as a difference of cubes. Let’s rewrite the exponents in a cubic form:
[ (x+h)^{\frac{2}{3}} - x^{\frac{2}{3}} = \left[ (x+h)^{\frac{1}{3}} \right]^2 - \left[ x^{\frac{1}{3}} \right]^2 ]
Now apply the difference of cubes identity:
Here, and , so:
[ (x+h)^{\frac{2}{3}} - x^{\frac{2}{3}} = \left[ (x+h)^{\frac{1}{3}} - x^{\frac{1}{3}} \right] \cdot \left[ (x+h)^{\frac{1}{3}} + x^{\frac{1}{3}} \right] ]
Step 3: Substitute back into the limit expression
Now, substitute this back into the limit:
[ f'(x) = 3 \lim_{h \to 0} \frac{\left[ (x+h)^{\frac{1}{3}} - x^{\frac{1}{3}} \right] \cdot \left[ (x+h)^{\frac{1}{3}} + x^{\frac{1}{3}} \right]}{h} ]
Step 4: Focus on simplifying the first factor
Now, consider only the first factor . This expression is a difference of two cube roots, and we can simplify it using the difference of cubes identity:
Let’s apply this formula with and :
Step 5: Substitute this into the limit expression
Now substitute this result back into the limit:
The terms cancel out:
Step 6: Take the limit as
As , , and the expression simplifies:
Final Answer:
Thus, the derivative of is:
Would you like more details or have any questions?
Here are five related questions you might find useful:
- How do you apply the difference of cubes in other situations?
- How can you use first principles to differentiate in general?
- Can you generalize the first-principles method for fractional powers?
- How does this compare to using standard derivative rules for power functions?
- What happens if the exponent in is negative or irrational?
Tip: When differentiating using first principles, breaking down complex terms (like roots or powers) using algebraic identities like the difference of cubes simplifies the process.
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Math Problem Analysis
Mathematical Concepts
Differentiation
First Principles
Algebra
Difference of Cubes
Formulas
f'(x) = lim(h -> 0) [f(x+h) - f(x)] / h
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
a^2 - b^2 = (a - b)(a + b)
Theorems
Difference of Cubes
Limit Definition of the Derivative
Suitable Grade Level
Grades 11-12 (Advanced Calculus)
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