Math Problem Statement
Solution
The function given is:
We are tasked with finding the derivative of this function. To do so, we will apply the product rule of differentiation and handle the cube root as a fractional exponent.
Step 1: Rewrite the function
First, express the cube root as a fractional exponent:
Step 2: Apply the Product Rule
The product rule states:
Here, and .
Differentiating :
Differentiating :
Step 3: Apply the Product Rule
Now substitute these into the product rule:
Step 4: Simplify the expression
You can leave the derivative in this form, or if needed, further simplify:
This is the derivative of the function.
Do you want more details on any part of this process, or have any questions?
Here are five related questions:
- How does the product rule apply when both terms are functions of ?
- How do you simplify expressions with fractional exponents?
- What is the difference between applying the product rule and the chain rule?
- How would the derivative change if the exponent were different?
- Can this derivative be applied to find the slope at a specific point?
Tip: When simplifying derivatives with fractional exponents, always rewrite them in forms that are easier to work with (e.g., as powers of instead of roots).
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Math Problem Analysis
Mathematical Concepts
Differentiation
Product Rule
Fractional Exponents
Formulas
Product Rule: (fg)' = f'g + fg'
Derivative of a power: d/dx[x^n] = n*x^(n-1)
Theorems
Product Rule of Differentiation
Suitable Grade Level
Grades 11-12