Let's solve the questions shown in the image step by step:

Question 2 (a)

We are asked to find the integral of:

$\int \frac{10x - 20}{x^2 - 4x + 5} \, dx$

Step-by-step solution:

1. First, notice that the numerator $10x - 20$ is the derivative of the denominator $x^2 - 4x + 5$, which suggests that a substitution method will work here.

2. Let: $u = x^2 - 4x + 5$ Then, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 2x - 4$

   Now, rewrite the integral: $\int \frac{10x - 20}{x^2 - 4x + 5} \, dx = \int \frac{10(x - 2)}{u} \, dx$

   But since $10(x - 2)$ is a constant multiple of $\frac{du}{dx}$, the integral simplifies: $\int \frac{du}{u} = \ln|u| + C = \ln|x^2 - 4x + 5| + C$ So, the final result is: $\int \frac{10x - 20}{x^2 - 4x + 5} \, dx = \ln|x^2 - 4x + 5| + C$

Question 2 (b)

We are given: $x^2y^2 + xy^2 = 4x$ and are asked to find $\frac{dy}{dx}$ when $x = 1$ and $y = 1$.

Step-by-step solution:

1. Differentiate both sides of the equation implicitly with respect to $x$: $\frac{d}{dx}(x^2 y^2 + xy^2) = \frac{d}{dx}(4x)$

2. Use the product rule and chain rule: $\frac{d}{dx}(x^2 y^2) = 2x y^2 + x^2(2y \frac{dy}{dx})$ and $\frac{d}{dx}(xy^2) = y^2 + x(2y \frac{dy}{dx})$

   So, the derivative becomes: $2x y^2 + 2x^2 y \frac{dy}{dx} + y^2 + 2xy \frac{dy}{dx} = 4$

3. Substitute $x = 1$ and $y = 1$ into the equation: $2(1)(1)^2 + 2(1)^2(1) \frac{dy}{dx} + (1)^2 + 2(1)(1) \frac{dy}{dx} = 4$ Simplifying: $2 + 2 \frac{dy}{dx} + 1 + 2 \frac{dy}{dx} = 4$ $3 + 4 \frac{dy}{dx} = 4$ $4 \frac{dy}{dx} = 1$ $\frac{dy}{dx} = \frac{1}{4}$

Question 2 (c)

Part (i)

We are given $y = x^{3x}$ and need to find $\frac{dy}{dx}$.

1. Take the natural logarithm of both sides to simplify: $\ln y = \ln(x^{3x}) = 3x \ln x$

2. Differentiate both sides using implicit differentiation: $\frac{1}{y} \frac{dy}{dx} = 3 \ln x + 3$

3. Multiply through by $y = x^{3x}$: $\frac{dy}{dx} = x^{3x} (3 \ln x + 3)$

Part (ii)

We are given $y = \tan^4(x)$ and need to find $\frac{dy}{dx}$.

1. Use the chain rule. First, rewrite the function: $y = (\tan x)^4$

2. Differentiate using the chain rule: $\frac{dy}{dx} = 4(\tan x)^3 \cdot \sec^2 x$

Do you want any further details or have any questions on any of the steps?

Here are 5 related questions to expand your understanding:

1. How do you identify when substitution is useful in integration?
2. What are some common applications of implicit differentiation in physics or engineering?
3. Can you generalize how to find the derivative of exponential functions like $y = x^{nx}$?
4. What is the difference between differentiating algebraic functions and trigonometric functions?
5. How can you apply the chain rule when differentiating composite functions?

Tip: In implicit differentiation, always apply the chain rule to terms involving $y$, as they are treated as functions of $x$.