https://file.aiquickdraw.com/mathbot/file/1729602966947vbt7zoi2.jpg

I see you've uploaded a math problem set. Let's go through the derivatives of the given functions.

### Derivatives:

1. **\( f(x) = \ln(2x^2 + 1) \)**:
   Using the chain rule:
   \[
   f'(x) = \frac{1}{2x^2 + 1} \cdot (4x) = \frac{4x}{2x^2 + 1}
   \]

2. **\( g(x) = \ln\left(\frac{5x^2}{x + 4}\right) \)**:
   Using logarithmic differentiation:
   \[
   g'(x) = \frac{1}{\frac{5x^2}{x + 4}} \cdot \frac{d}{dx}\left(\frac{5x^2}{x + 4}\right)
   \]
   First, differentiate \( \frac{5x^2}{x + 4} \) using the quotient rule:
   \[
   \frac{d}{dx} \left(\frac{5x^2}{x+4}\right) = \frac{(10x)(x + 4) - (5x^2)(1)}{(x+4)^2} = \frac{10x(x + 4) - 5x^2}{(x + 4)^2}
   = \frac{5x(2x + 8)}{(x + 4)^2}
   \]
   So:
   \[
   g'(x) = \frac{(x + 4)^2}{5x^2} \cdot \frac{10x(x + 4)}{(x + 4)^2} = \frac{10x(x + 4)}{5x^2}
   \]

3. **\( h(x) = e^{-3/x^2} \)**:
   Using the chain rule:
   \[
   h'(x) = e^{-3/x^2} \cdot \frac{d}{dx}\left(-\frac{3}{x^2}\right) = e^{-3/x^2} \cdot \frac{6}{x^3}
   \]

4. **\( j(x) = xe^{-x} \)**:
   Using the product rule:
   \[
   j'(x) = e^{-x} + x(-e^{-x}) = e^{-x}(1 - x)
   \]

5. **\( k(x) = 7 - 2x \)**:
   The derivative is straightforward:
   \[
   k'(x) = -2
   \]

6. **\( m(x) = \log_3 \left(\frac{3x^2 + 1}{x}\right) \)**:
   Using the chain rule and logarithmic differentiation:
   \[
   m'(x) = \frac{1}{\ln(3)} \cdot \frac{d}{dx} \left(\frac{3x^2 + 1}{x}\right)
   \]
   Differentiate \( \frac{3x^2 + 1}{x} \) using the quotient rule:
   \[
   \frac{d}{dx} \left(\frac{3x^2 + 1}{x}\right) = \frac{(6x)(x) - (3x^2 + 1)(1)}{x^2} = \frac{6x^2 - (3x^2 + 1)}{x^2} = \frac{3x^2 - 1}{x^2}
   \]
   So:
   \[
   m'(x) = \frac{1}{\ln(3)} \cdot \frac{3x^2 - 1}{x^2}
   \]

---

### Additional Problems:

The next few questions involve finding equations of tangent lines. To find those, you'll first need to compute the derivative of the given function and then evaluate it at the specified point to get the slope of the tangent line.

Would you like to dive deeper into any specific derivative or equation of the tangent line?

Here are 5 follow-up questions to consider:

1. Do you need help solving the tangent line problems starting from question 11?
2. Would you like a step-by-step explanation of the quotient rule used in problem 2?
3. Are there any specific derivative rules you want clarification on (e.g., chain rule, product rule)?
4. Should I go over implicit differentiation for problems 7-10?
5. Would you like me to graph one of these functions to illustrate the tangent line concept visually?

**Tip**: When applying the chain rule, always remember to differentiate the outer function first, then multiply by the derivative of the inner function.

Find the derivatives and solve the tangent line problems as listed in the image.

Solve derivatives of logarithmic, exponential, and product functions using chain, product, and quotient rules. Find tangent lines for functions involving trigonometric and implicit differentiation. Perfect for advanced high school or undergraduate 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