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The given expression asks for the derivative of \( x^3 - \cos(x) \) with respect to \( x \). Let's break it down step by step.

The function to differentiate is:

\[
f(x) = x^3 - \cos(x)
\]

### Step 1: Differentiate \( x^3 \)
The derivative of \( x^3 \) with respect to \( x \) is found using the power rule:

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

### Step 2: Differentiate \( -\cos(x) \)
The derivative of \( -\cos(x) \) with respect to \( x \) is:

\[
\frac{d}{dx} (-\cos(x)) = \sin(x)
\]

### Step 3: Combine the derivatives
Now, sum the results from Step 1 and Step 2:

\[
\frac{d}{dx} \left( x^3 - \cos(x) \right) = 3x^2 + \sin(x)
\]

### Final Answer:
\[
\frac{d}{dx} \left( x^3 - \cos(x) \right) = 3x^2 + \sin(x)
\]

Let me know if you'd like a more detailed explanation or have any questions!

Here are five follow-up questions related to derivatives:
1. What is the derivative of \( \sin(x) \) and how does it differ from \( \cos(x) \)?
2. How do you apply the product rule for derivatives in calculus?
3. Can you find the second derivative of \( x^3 - \cos(x) \)?
4. How does the chain rule apply to more complex functions, like \( \cos(x^2) \)?
5. How would you compute the derivative of a function involving both \( e^x \) and trigonometric functions?

**Tip**: Remember to apply basic differentiation rules like the power rule, chain rule, and trigonometric derivatives accurately for more complex functions!

What is the derivative of x^3 - cos(x) with respect to x?

Find the derivative of x^3 - cos(x) using the power rule and trigonometric differentiation. This detailed solution walks through each step, providing a clear understanding of how to apply basic calculus rules. Suitable for high school students studying Calculus.

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