The problem in the image asks to determine $f'(x)$, the derivative of the function $f(x) = \frac{2}{3}x^3 - \frac{3}{4}x^2 + 4$.

Solution:

The derivative $f'(x)$ is calculated term by term:

1. $\frac{d}{dx} \left( \frac{2}{3}x^3 \right) = \frac{2}{3} \cdot 3x^{3-1} = 2x^2$
2. $\frac{d}{dx} \left( -\frac{3}{4}x^2 \right) = -\frac{3}{4} \cdot 2x^{2-1} = -\frac{3}{2}x$
3. $\frac{d}{dx} \left( 4 \right) = 0$ (the derivative of a constant is 0)

Now, combine the terms: $f'(x) = 2x^2 - \frac{3}{2}x$

Final Answer:

$f'(x) = 2x^2 - \frac{3}{2}x$

Would you like me to go into more detail about any of the steps?

Related Questions:

1. How do you calculate higher-order derivatives, such as $f''(x)$?
2. How does the derivative change if the coefficients are functions of $x$ (e.g., $f(x) = g(x)x^3$)?
3. What is the significance of the derivative in practical applications, such as physics or economics?
4. Can you explain the role of each term in determining the behavior of $f(x)$ and $f'(x)$?
5. How would the derivative differ if the function had trigonometric terms, e.g., $f(x) = \sin(x) + \frac{2}{3}x^3$?

Tip:

Always simplify the function before differentiating to avoid unnecessary complications!