https://file.aiquickdraw.com/mathbot/file/1726009064688cdp6ykwh.jpg

In this image, the mathematical problem involves two functions \( u(x) \) and \( v(x) \), with a relation between them given as:

\[
u(x) = (f(x))^2
\]
\[
v(x) = f(x^2)
\]

The task requests:

1. To express \( u'(x) \) in terms of \( f(x) \) and \( f'(x) \) using the chain rule.
2. To use the chain rule to find the derivative of the composite function for \( v(x) \).

### Solution:

#### 1. Finding \( u'(x) \):
We are given that \( u(x) = (f(x))^2 \). To differentiate \( u(x) \), we apply the chain rule. The derivative of \( u(x) \) is:

\[
u'(x) = 2 f(x) f'(x)
\]

#### 2. Finding \( v'(x) \):
For \( v(x) = f(x^2) \), we again apply the chain rule. The outer function is \( f \) and the inner function is \( x^2 \). So the derivative is:

\[
v'(x) = f'(x^2) \cdot 2x
\]

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If you have further questions or need more details, feel free to ask!

Here are 5 related questions to deepen your understanding:
1. What is the chain rule, and how does it apply to composite functions?
2. Can you generalize the derivative of a product of two functions using the product rule?
3. What is the difference between differentiating a squared function and differentiating a composite function?
4. How do higher-order derivatives apply to composite functions like \( v(x) \)?
5. How does implicit differentiation compare to the standard differentiation seen here?

**Tip:** Always apply the chain rule carefully when differentiating composite functions, ensuring you account for both the inner and outer functions.

The mathematical problem involves two functions u(x) and v(x), defined as: u(x) = (f(x))^2 and v(x) = f(x^2). The tasks involve finding u'(x) and v'(x) using the chain rule.

Learn how to differentiate composite functions using the chain rule. This problem covers finding the derivatives of u(x) = (f(x))^2 and v(x) = f(x^2). Detailed step-by-step solutions are provided. Ideal for university-level calculus students.

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