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The problem asks you to find the derivatives of different combinations of functions, where $f(x)$ is a differentiable function of $x$. Here's the breakdown of the given subproblems:

1. Part a: $\frac{d}{dx} \left[ x^2 + f(x) \right]$

   • The derivative of $x^2$ is $2x$.
   • The derivative of $f(x)$ is $f'(x)$.
   • So, the correct derivative is:
     $\frac{d}{dx} \left[ x^2 + f(x) \right] = 2x + f'(x)$

2. Part b: $\frac{d}{dx} \left[ x^2 f(x) \right]$

   • Use the product rule: $\frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)$.
   • Here, $u(x) = x^2$ and $v(x) = f(x)$.
   • The derivative of $x^2$ is $2x$, and the derivative of $f(x)$ is $f'(x)$.
   • So, the correct derivative is: $\frac{d}{dx} \left[ x^2 f(x) \right] = 2x f(x) + x^2 f'(x)$

3. Part c: $\frac{d}{dx} \left[ c + x + f(x)^2 \right]$

   • The derivative of $c$ (a constant) is 0.
   • The derivative of $x$ is 1.
   • The derivative of $f(x)^2$ is $2 f(x) f'(x)$ (using the chain rule).
   • So, the correct derivative is: $\frac{d}{dx} \left[ c + x + f(x)^2 \right] = 1 + 2 f(x) f'(x)$

4. Part d: $\frac{d}{dx} \left[ f(x)^2 \right]$

   • Use the chain rule: the derivative of $f(x)^2$ is $2 f(x) f'(x)$.
   • So, the correct derivative is: $\frac{d}{dx} \left[ f(x)^2 \right] = 2 f(x) f'(x)$

5. Part e: $\frac{d}{dx} \left[ x f(x) + f(cx) \right]$

   • Use the product rule for $x f(x)$, which gives $f(x) + x f'(x)$.
   • Use the chain rule for $f(cx)$: the derivative of $f(cx)$ is $f'(cx) \cdot c$ (since the derivative of $cx$ is $c$).
   • So, the correct derivative is: $\frac{d}{dx} \left[ x f(x) + f(cx) \right] = f(x) + x f'(x) + c f'(cx)$

Would you like to check the results in more detail or do you have any further questions on this? Here are some related topics you can explore:

1. What is the chain rule, and how does it apply to derivatives of compositions of functions?
2. How does the product rule work for differentiating products of functions?
3. What are some other derivative rules commonly used in calculus?
4. How do you find the derivatives of higher-order polynomials with mixed functions?
5. Can the results in these problems be applied to integrals?

Tip: Always double-check your work when using the product rule or chain rule, as small mistakes can lead to incorrect answers.