Let's solve the derivatives for each of the given functions step by step.

1. Function 1: $y = (tnz^2 + 6x)^2$

• Derivative: $\text{Let } u = tnz^2 + 6x.$ $y = u^2$ The derivative of $y$ with respect to $x$ is: $\frac{dy}{dx} = 2u \cdot \frac{du}{dx}$ Now, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 2tnz \cdot \frac{dz}{dx} + 6$ Substituting back: $\frac{dy}{dx} = 2(tnz^2 + 6x) \cdot (2tnz \cdot \frac{dz}{dx} + 6)$

2. Function 2: $y = 2x^{2x+5}$

• Derivative: We use logarithmic differentiation: $\ln y = \ln(2x^{2x+5}) = \ln 2 + (2x + 5) \ln x$ Differentiating both sides with respect to $x$: $\frac{1}{y} \cdot \frac{dy}{dx} = 0 + \left( 2 \cdot \ln x + \frac{2x + 5}{x} \right)$ So, $\frac{dy}{dx} = y \cdot \left( 2 \ln x + \frac{2x + 5}{x} \right)$ Substitute back $y = 2x^{2x+5}$: $\frac{dy}{dx} = 2x^{2x+5} \cdot \left( 2 \ln x + \frac{2x + 5}{x} \right)$

3. Function 3: $y = \frac{4x^2 + 2}{x^6}$

• Derivative: Simplify first: $y = \frac{4x^2}{x^6} + \frac{2}{x^6} = 4x^{-4} + 2x^{-6}$ Differentiate term by term: $\frac{dy}{dx} = -16x^{-5} - 12x^{-7} = -\frac{16}{x^5} - \frac{12}{x^7}$

4. Function 4: $y = (5x + 7)(3x^2 + 5)$

• Derivative: Use the product rule: [ \frac{dy}{dx} = \frac{d}{dx}[5x + 7] \cdot (3x^2 + 5) + (5x + 7) \cdot \frac{d}{dx}[3x^2 + 5] ] Differentiate each: $\frac{dy}{dx} = 5(3x^2 + 5) + (5x + 7)(6x)$ Simplify: $\frac{dy}{dx} = 15x^2 + 25 + 30x^2 + 42x = 45x^2 + 42x + 25$

Questions:

1. How do you apply the product rule in differentiation?
2. What is logarithmic differentiation and when is it useful?
3. Why is it important to simplify functions before differentiating?
4. How does the chain rule work in composite functions?
5. What are some common mistakes to avoid when differentiating complex functions?

Tip: Always check if simplifying a function first makes the differentiation process easier.