https://file.aiquickdraw.com/mathbot/file/1726715300489xi7qttmn.jpg

The function given in the image is:

\[
f(x) = -3x^8
\]

We are asked to find the derivative, \( f'(x) \), using the rules of differentiation.

To differentiate this function, we apply the **power rule**, which states:

\[
\frac{d}{dx}(x^n) = n \cdot x^{n-1}
\]

Here, the constant multiple rule also applies, where the derivative of a constant times a function is the constant times the derivative of the function.

Now, differentiating \( f(x) = -3x^8 \):

1. The constant \(-3\) remains as is.
2. Apply the power rule to \( x^8 \), giving \( 8x^{7} \).

Thus, the derivative is:

\[
f'(x) = -3 \cdot 8x^7 = -24x^7
\]

So the answer is:

\[
f'(x) = -24x^7
\]

Would you like more details, or do you have any questions?

Here are 5 related questions for practice:
1. What is the derivative of \( f(x) = 5x^9 \)?
2. Differentiate \( f(x) = -2x^6 + 4x^3 \).
3. What is the second derivative of \( f(x) = -3x^8 \)?
4. Find the derivative of \( f(x) = 7x^{10} - 3x^5 \).
5. How does the power rule apply to negative exponents like \( f(x) = 4x^{-2} \)?

**Tip:** Always apply the constant multiple rule when differentiating functions that involve a constant multiplied by a variable term!

Find the derivative of the function f(x) = -3x^8 using the rules of differentiation.

Find the derivative of f(x) = -3x^8 by applying the power rule of differentiation. This solution explains step-by-step how to calculate the derivative, including the use of the constant multiple rule. Suitable for high school calculus students in Grades 11-12.

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