Find the derivative of f left parenthesis x right parenthesis equals x Superscript 7
.

The given function is \( f(x) = x^7 \).

To find the derivative of this function, we apply the power rule. The power rule states that for a function \( f(x) = x^n \), its derivative is given by:

\[
f'(x) = n \cdot x^{n-1}
\]

For \( f(x) = x^7 \), we can apply this rule:

\[
f'(x) = 7 \cdot x^{7-1} = 7x^6
\]

Thus, the derivative of \( f(x) = x^7 \) is:

\[
f'(x) = 7x^6
\]

Would you like further details on this process or have any questions?

Here are some related questions:
1. What is the derivative of \( f(x) = x^5 \)?
2. How do you apply the power rule to a function like \( f(x) = x^{10} \)?
3. What happens if the exponent is a fraction, like in \( f(x) = x^{1/2} \)?
4. How would you differentiate \( f(x) = x^7 + 3x^4 \)?
5. Can you explain the product rule in differentiation?

**Tip:** Remember that the power rule applies only when the variable has a constant exponent!

Learn how to find the derivative of f(x) = x^7 using the power rule in calculus. This step-by-step guide explains the application of the power rule formula f'(x) = n * x^(n-1) and is ideal for students in Grades 11-12 and college-level calculus.

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