I see you've uploaded an image containing mathematical problems requiring differentiation. I'll solve each question step by step.

(a) $f(x) = 10^{2x - x^2} + x \ln x^2$

1. Use the chain rule and logarithmic differentiation to find $\frac{d}{dx}$.
   • First term: $\frac{d}{dx}[10^{2x - x^2}] = 10^{2x - x^2} \cdot \ln(10) \cdot \frac{d}{dx}(2x - x^2) = 10^{2x - x^2} \cdot \ln(10) \cdot (2 - 2x)$.
   • Second term: $\frac{d}{dx}[x \ln x^2] = \frac{d}{dx}[x \cdot 2 \ln x] = \frac{d}{dx}[2x \ln x] = 2 \ln x + 2$.

Final Answer: $\frac{df}{dx} = 10^{2x - x^2} \ln(10)(2 - 2x) + 2 \ln x + 2$

(b) Parametric differentiation: $x = t^2 + 4t$, $y = \tan t$

1. Differentiate $x$ and $y$ with respect to $t$:

   • $\frac{dx}{dt} = 2t + 4$
   • $\frac{dy}{dt} = \sec^2 t$

2. Use $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$:

   • $\frac{dy}{dx} = \frac{\sec^2 t}{2t + 4}$.

Final Answer: $\frac{dy}{dx} = \frac{\sec^2 t}{2t + 4}$

(c) $y = \log(u^2 - 2)$, $u = \sqrt{x}$

1. Substitute $u = \sqrt{x}$: $y = \log((\sqrt{x})^2 - 2) = \log(x - 2)$.

2. Differentiate $y$ with respect to $x$:

   • $\frac{dy}{dx} = \frac{1}{x - 2} \cdot \frac{d}{dx}[x - 2] = \frac{1}{x - 2}$.

Final Answer: $\frac{dy}{dx} = \frac{1}{x - 2}$

(d) $g(x) = f^{-1}(x)$, where $f(x) = e^{2x}$

1. For inverse functions: $f(g(x)) = x$.

   • Differentiate both sides: $f'(g(x)) \cdot g'(x) = 1$.

2. Compute $f'(x) = \frac{d}{dx}[e^{2x}] = 2e^{2x}$.

3. Substituting $f'(g(x)) = 2e^{2g(x)}$:

   • $g'(x) = \frac{1}{2e^{2g(x)}}$.

4. Since $g(x)$ is the inverse, $g(x) = \frac{1}{2} \ln(x)$:

   • $g'(x) = \frac{1}{2x}$.

Final Answer: $g'(x) = \frac{1}{2x}$

Let me know if you'd like further explanations or examples. Here are 5 questions for deeper exploration:

1. How does the chain rule apply to logarithmic differentiation in part (a)?
2. Why is parametric differentiation necessary in part (b)?
3. Could you generalize part (c) for $u$ being any function of $x$?
4. How does the derivative of an inverse function relate to its original function in part (d)?
5. What are the steps to simplify expressions with logarithmic or exponential functions?

Tip: Always rewrite functions in a simpler form before differentiating to avoid errors!