To find the gradient of the given function $y = 5 + 8x - \frac{4}{3}x^3$ at $x = -1$, we need to differentiate the function with respect to $x$ and then evaluate the derivative at $x = -1$.

Step 1: Differentiate the function

The function is $y = 5 + 8x - \frac{4}{3}x^3$. We will differentiate each term:

• The derivative of $5$ is $0$.
• The derivative of $8x$ is $8$.
• The derivative of $-\frac{4}{3}x^3$ is $-\frac{4}{3} \cdot 3x^2 = -4x^2$.

So, the derivative (gradient function) is: $\frac{dy}{dx} = 8 - 4x^2$

Step 2: Evaluate the derivative at $x = -1$

Now, substitute $x = -1$ into the derivative: $\frac{dy}{dx} = 8 - 4(-1)^2 = 8 - 4(1) = 8 - 4 = 4$

Final Answer:

The gradient of the function at $x = -1$ is $4$.

Would you like more details on the differentiation steps or have any questions? Here are some related questions:

1. How do we differentiate more complex functions like trigonometric or logarithmic expressions?
2. What does the gradient of a function tell us about its behavior at a specific point?
3. How can the power rule for differentiation be extended to fractional powers of $x$?
4. How does differentiation work when we deal with products or quotients of functions?
5. What is the significance of second derivatives, and how do they relate to concavity?

Tip: To find the gradient of a curve at any point, always remember to differentiate first, and then substitute the specific $x$-value into the derivative!