https://file.aiquickdraw.com/mathbot/file/1735644806125gx5qpn4a.jpg

The image contains four functions, and the task is to differentiate them using derivative rules. Let’s compute the derivatives for each function step by step:

---

### 1) \( f(x) = \frac{1}{f} x^f - x^f \)

The derivative is computed as follows:
\[
f'(x) = \frac{1}{f} \cdot f x^{f-1} - f x^{f-1}
\]
Simplify:
\[
f'(x) = x^{f-1} \left(\frac{1}{f} \cdot f - f\right) = x^{f-1} \left(1 - f\right)
\]

---

### 2) \( v(r) = \frac{f}{r} \pi r^f \)

This can be rewritten as:
\[
v(r) = f \pi r^{f-1}
\]
Taking the derivative:
\[
v'(r) = f \pi (f-1) r^{f-2}
\]

---

### 3) \( s(r) = \frac{1}{r} (\pi r^2 + 1 \cdot \pi r) \)

Simplify the function first:
\[
s(r) = \frac{\pi r^2}{r} + \frac{\pi r}{r} = \pi r + \frac{\pi}{r}
\]
Take the derivative term by term:
\[
s'(r) = \pi - \frac{\pi}{r^2}
\]

---

### 4) \( f(x) = \frac{x^f}{f} + \frac{r}{x^r} \)

The derivative is:
\[
f'(x) = \frac{1}{f} \cdot f x^{f-1} + \frac{d}{dx} \left(\frac{r}{x^r}\right)
\]
For the second term, use the power rule:
\[
\frac{d}{dx} \left(\frac{r}{x^r}\right) = r \cdot (-r) x^{-r-1} = -r^2 x^{-r-1}
\]
Thus:
\[
f'(x) = x^{f-1} - r^2 x^{-r-1}
\]

---

### Final Results:

1. \( f'(x) = x^{f-1} (1 - f) \)
2. \( v'(r) = f \pi (f-1) r^{f-2} \)
3. \( s'(r) = \pi - \frac{\pi}{r^2} \)
4. \( f'(x) = x^{f-1} - r^2 x^{-r-1} \)

---

Do you have any questions, or would you like further clarifications?  

### Related Questions:
1. What are the steps to simplify complex derivative expressions like these?
2. How can the chain rule be applied in multivariable functions?
3. Why is the power rule fundamental in differentiating polynomial-like terms?
4. How would these derivatives change if additional terms were added?
5. Can these rules be generalized for higher derivatives?

### Tip:
Always simplify the function as much as possible before differentiating—it reduces errors and makes computation easier!

Differentiate the given functions step by step as shown in the image.

Explore how to differentiate complex functions using derivative rules like the power rule and quotient rule. Includes step-by-step solutions for multiple examples involving simplification and application of calculus concepts. Ideal for calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation