Find the first and second derivatives of the function.
f(x) = (3x2 + 8)7/2

We are given the function:

\[
f(x) = \left(3x^2 + 8\right)^{7/2}
\]

### First Derivative

To find the first derivative \( f'(x) \), we can apply the **chain rule**. The chain rule is used when differentiating a composite function, which is the case here. The outer function is \( u^{7/2} \) and the inner function is \( u = 3x^2 + 8 \).

Step 1: Differentiate the outer function \( u^{7/2} \) with respect to \( u \):

\[
\frac{d}{du}\left(u^{7/2}\right) = \frac{7}{2} u^{5/2}
\]

Step 2: Differentiate the inner function \( u = 3x^2 + 8 \) with respect to \( x \):

\[
\frac{d}{dx}(3x^2 + 8) = 6x
\]

Now, applying the chain rule:

\[
f'(x) = \frac{7}{2} \left(3x^2 + 8\right)^{5/2} \cdot 6x
\]

Simplifying:

\[
f'(x) = 21x \left(3x^2 + 8\right)^{5/2}
\]

### Second Derivative

To find the second derivative \( f''(x) \), we apply the product rule and chain rule to \( f'(x) = 21x \left(3x^2 + 8\right)^{5/2} \).

Using the product rule:

\[
f''(x) = \frac{d}{dx}\left(21x\right) \cdot \left(3x^2 + 8\right)^{5/2} + 21x \cdot \frac{d}{dx}\left( \left(3x^2 + 8\right)^{5/2} \right)
\]

Step 1: Differentiate \( 21x \):

\[
\frac{d}{dx}(21x) = 21
\]

Step 2: Differentiate \( \left(3x^2 + 8\right)^{5/2} \) using the chain rule again:

\[
\frac{d}{dx} \left( \left(3x^2 + 8\right)^{5/2} \right) = \frac{5}{2} \left(3x^2 + 8\right)^{3/2} \cdot 6x
\]

Simplifying:

\[
\frac{d}{dx} \left( \left(3x^2 + 8\right)^{5/2} \right) = 15x \left(3x^2 + 8\right)^{3/2}
\]

Now substituting back into the product rule:

\[
f''(x) = 21 \left(3x^2 + 8\right)^{5/2} + 21x \cdot 15x \left(3x^2 + 8\right)^{3/2}
\]

Simplifying further:

\[
f''(x) = 21 \left(3x^2 + 8\right)^{5/2} + 315x^2 \left(3x^2 + 8\right)^{3/2}
\]

Thus, the second derivative is:

\[
f''(x) = 21 \left(3x^2 + 8\right)^{5/2} + 315x^2 \left(3x^2 + 8\right)^{3/2}
\]

Let me know if you'd like more details on any part of the process or have any questions!

### 5 Related Questions:
1. What is the physical interpretation of the first and second derivatives in real-world applications?
2. How do we apply the chain rule in more complex multi-variable functions?
3. Can the second derivative be used to determine concavity and points of inflection for this function?
4. How can we generalize the chain rule for higher-order derivatives?
5. How would the result change if the exponent of the function were different?

### Tip:
Always apply the chain rule carefully when differentiating composite functions, ensuring you differentiate both the outer and inner functions correctly.

Find the first and second derivatives of the function f(x) = (3x^2 + 8)^(7/2).

Learn how to find the first and second derivatives of the function f(x) = (3x^2 + 8)^(7/2) step by step. This tutorial covers the application of the chain rule and product rule, key concepts in calculus, and is suitable for students in Grades 11-12.

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