https://r2.erweima.ai/mathbot/image/1726511151618xu2zjv3k.jpg

We are asked to compute the derivative of the function given as:

\[
f(x) = \frac{\sqrt{x} \cdot \sqrt[3]{3x}}{2^5 \cdot \sqrt[3]{3x^2}} \cdot e^4
\]

Let's simplify the function first:

1. The term \( e^4 \) is constant, so we can factor it out and ignore it during the differentiation process.

2. Simplifying the radicals:
   - \( \sqrt{x} = x^{1/2} \)
   - \( \sqrt[3]{3x} = (3x)^{1/3} = 3^{1/3} \cdot x^{1/3} \)
   - \( \sqrt[3]{3x^2} = (3x^2)^{1/3} = 3^{1/3} \cdot x^{2/3} \)

Thus, the function becomes:

\[
f(x) = e^4 \cdot \frac{x^{1/2} \cdot 3^{1/3} \cdot x^{1/3}}{2^5 \cdot 3^{1/3} \cdot x^{2/3}}
\]

The terms \( 3^{1/3} \) cancel out. Now simplify the powers of \( x \):

\[
f(x) = e^4 \cdot \frac{x^{1/2 + 1/3}}{2^5 \cdot x^{2/3}} = e^4 \cdot \frac{x^{5/6}}{32 \cdot x^{2/3}}
\]

Now, subtract the exponents of \( x \):

\[
f(x) = e^4 \cdot \frac{x^{5/6 - 2/3}}{32} = e^4 \cdot \frac{x^{1/6}}{32}
\]

Thus, the simplified function is:

\[
f(x) = \frac{e^4}{32} \cdot x^{1/6}
\]

### Derivative of the function:

The derivative of \( f(x) = C \cdot x^{1/6} \) (where \( C = \frac{e^4}{32} \)) is given by:

\[
f'(x) = C \cdot \frac{1}{6} \cdot x^{-5/6}
\]

Substitute \( C \):

\[
f'(x) = \frac{e^4}{32} \cdot \frac{1}{6} \cdot x^{-5/6} = \frac{e^4}{192} \cdot x^{-5/6}
\]

This is the derivative of the function. 

Would you like further details or clarification on any step?

---

Here are 5 related questions you might want to explore:
1. What is the derivative of a general power function \( f(x) = x^n \)?
2. How do you differentiate products and quotients involving constants like \( e^4 \)?
3. What is the process for differentiating functions with radicals?
4. How do you simplify expressions with fractional exponents before taking derivatives?
5. What is the role of constant factors like \( e^4 \) in differentiation?

**Tip**: When simplifying expressions with radicals, converting them to fractional exponents makes differentiation much easier.

Calcula la siguiente derivada de la siguiente función: f(x) = (sqrt(x) * sqrt[3](3x)) / (2^5 * sqrt[3](3x^2)) * e^4

Learn how to find the derivative of the function f(x) = (sqrt(x) * sqrt[3](3x)) / (2^5 * sqrt[3](3x^2)) * e^4 by simplifying expressions with radicals and applying the power rule. Suitable for undergraduate 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