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Here are the differential coefficients for the given expressions:

1. For \( \left( \tan^2 z \right)^{2 \ln z} \), the derivative with respect to \( z \) is:
\[
\frac{d}{dz} \left( \left( \tan^2 z \right)^{2 \ln z} \right) = \left( 2(2\tan^2 z + 2) \frac{\ln z}{\tan z} + \frac{2 \ln(\tan^2 z)}{z} \right) \left( \tan^2 z \right)^{2 \ln z}
\]

2. For \( \sqrt{x} \sin^2(3x^3 - 5) \), the derivative with respect to \( x \) is:
\[
\frac{d}{dx} \left( \sqrt{x} \sin^2(3x^3 - 5) \right) = 18 x^{\frac{5}{2}} \sin(3x^3 - 5) \cos(3x^3 - 5) + \frac{\sin^2(3x^3 - 5)}{2\sqrt{x}}
\]

Would you like further details on the steps involved in deriving these? Here are some questions to explore the topic further:

1. How does logarithmic differentiation help simplify the first derivative?
2. Can you apply the chain rule to other trigonometric functions in a similar way?
3. What role does the product rule play in differentiating the second expression?
4. How do trigonometric identities affect the simplification of these derivatives?
5. How does the power of a trigonometric function influence its derivative?

**Tip:** When differentiating composite functions, it's crucial to apply the chain rule, especially for powers of trigonometric functions.

Find the differential coefficients of the followings: i. (tan^2 z)^(2 ln z) ii. √x sin^2(3x^3 - 5)

Explore the step-by-step process of differentiating complex expressions like (tan^2 z)^(2 ln z) and √x sin^2(3x^3 - 5). Learn the application of chain rule, product rule, and logarithmic differentiation in advanced calculus.

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