We are given the function $f(x) = x^3$ and are asked to compute both $f(-5)$ and $f'(-5)$.

1. Compute $f(-5)$

To compute $f(-5)$, we simply substitute $x = -5$ into the function $f(x) = x^3$:

$f(-5) = (-5)^3 = -125$

2. Compute $f'(x)$

To find the derivative $f'(x)$, we differentiate $f(x) = x^3$ using basic differentiation rules:

$f'(x) = \frac{d}{dx} \left( x^3 \right) = 3x^2$

Now, to compute $f'(-5)$, we substitute $x = -5$ into the derivative:

$f'(-5) = 3(-5)^2 = 3 \times 25 = 75$

Final Answers:

• $f(-5) = -125$
• $f'(-5) = 75$

Let me know if you need further explanation or details!

Related Questions:

1. How do you compute higher-order derivatives of $f(x) = x^3$?
2. What does the value of $f'(x)$ represent geometrically?
3. How does the derivative of a cubic function behave at different points?
4. How can you use the chain rule to differentiate composite functions like $f(g(x))$?
5. What are some common applications of derivatives in real-world scenarios?

Tip:

For polynomials like $x^3$, the power rule $\frac{d}{dx} x^n = n \cdot x^{n-1}$ is a quick way to find derivatives.