Let \displaystyle f(x)= 3 x^{2/5} - 6 x^{3/7}. Evaluate each of the following:


1.  	f'(2)	=	
2.  	f'(4)	=

To evaluate \( f'(x) \) at specific points, we first need to differentiate the function \( f(x) = 3x^{2/5} - 6x^{3/7} \).

### Step 1: Differentiate \( f(x) \)

We use the power rule for differentiation. The power rule states:

\[
\frac{d}{dx}\left( x^n \right) = n x^{n-1}
\]

#### For the first term, \( 3x^{2/5} \):

\[
\frac{d}{dx}\left( 3x^{2/5} \right) = 3 \cdot \frac{2}{5} \cdot x^{2/5 - 1} = \frac{6}{5} x^{-3/5}
\]

#### For the second term, \( -6x^{3/7} \):

\[
\frac{d}{dx}\left( -6x^{3/7} \right) = -6 \cdot \frac{3}{7} \cdot x^{3/7 - 1} = -\frac{18}{7} x^{-4/7}
\]

Thus, the derivative is:

\[
f'(x) = \frac{6}{5} x^{-3/5} - \frac{18}{7} x^{-4/7}
\]

### Step 2: Evaluate \( f'(x) \) at \( x = 2 \) and \( x = 4 \)

#### 1. \( f'(2) \):

\[
f'(2) = \frac{6}{5} \cdot 2^{-3/5} - \frac{18}{7} \cdot 2^{-4/7}
\]

Using approximate values for powers of 2:

\[
2^{-3/5} \approx 0.72478, \quad 2^{-4/7} \approx 0.75786
\]

Substituting these into the equation:

\[
f'(2) \approx \frac{6}{5} \cdot 0.72478 - \frac{18}{7} \cdot 0.75786
\]

\[
f'(2) \approx 0.869736 - 1.94843 = -1.078694
\]

So, \( f'(2) \approx -1.079 \).

#### 2. \( f'(4) \):

\[
f'(4) = \frac{6}{5} \cdot 4^{-3/5} - \frac{18}{7} \cdot 4^{-4/7}
\]

Using approximate values for powers of 4:

\[
4^{-3/5} \approx 0.54180, \quad 4^{-4/7} \approx 0.57435
\]

Substituting these into the equation:

\[
f'(4) \approx \frac{6}{5} \cdot 0.54180 - \frac{18}{7} \cdot 0.57435
\]

\[
f'(4) \approx 0.65016 - 1.47690 = -0.82674
\]

So, \( f'(4) \approx -0.827 \).

### Final Answers:
1. \( f'(2) \approx -1.079 \)
2. \( f'(4) \approx -0.827 \)

---

Would you like any details on the differentiation process or further explanations? 

Here are 5 related questions to explore:

1. What is the interpretation of a negative derivative in this context?
2. How does \( f'(x) \) behave as \( x \) approaches zero?
3. What are the critical points of \( f(x) \)?
4. Can you find the second derivative \( f''(x) \)?
5. How would the function change if \( f(x) \) were raised to a higher power?

**Tip:** Remember that fractional exponents follow the same differentiation rules as integer exponents—just apply the power rule carefully!

Let f(x)= 3 x^{2/5} - 6 x^{3/7}. Evaluate each of the following:

1. f'(2) =
2. f'(4) =

Learn how to differentiate the function f(x) = 3x^(2/5) - 6x^(3/7) using the power rule and evaluate the derivative at specific points, f'(2) and f'(4). This step-by-step solution explains key calculus concepts, including fractional exponents and differentiation techniques. Suitable for students in Grades 11-12 or college level.

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