Let us solve each of the problems one by one.

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1. Find $y'(x)$, where:

$y(x) = \frac{8 + 4x^2 + 3x^4}{15} \cdot \sqrt[3]{1 - x}.$

Solution:

We apply the product rule and chain rule: Let: $u(x) = \frac{8 + 4x^2 + 3x^4}{15}, \quad v(x) = \sqrt[3]{1 - x}.$

Derivatives:

1. $u'(x) = \frac{d}{dx} \left( \frac{8 + 4x^2 + 3x^4}{15} \right) = \frac{1}{15}(8x + 12x^3)$.
2. $v'(x) = \frac{d}{dx} \left( (1 - x)^{1/3} \right) = \frac{-1}{3}(1 - x)^{-2/3}$.

Product Rule:

$y'(x) = u'(x)v(x) + u(x)v'(x).$

Substitute $u(x), u'(x), v(x), v'(x)$ to find $y'(x)$.

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2. Find $y'(x)$, if:

$x = t^2 + 3t, \quad y = t^4 + 2t^3.$

Solution:

We find $\frac{dy}{dx}$ using the chain rule: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.$

Derivatives:

1. $\frac{dx}{dt} = 2t + 3$.
2. $\frac{dy}{dt} = 4t^3 + 6t^2$.

Substitute to get: $\frac{dy}{dx} = \frac{4t^3 + 6t^2}{2t + 3}.$

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3. Find $y''(x)$, if:

$y = (1 + x^2) \arctan(x).$

Solution:

We use the product rule and then differentiate again: $y' = \frac{d}{dx} \left( (1 + x^2) \arctan(x) \right) = \arctan(x) \cdot 2x + (1 + x^2) \cdot \frac{1}{1 + x^2}.$

Simplify $y'$, and then differentiate again to get $y''$.

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4. Find $\int \frac{\sin(x)}{\cos(x) + 4} \, dx$.

Solution:

We use substitution: Let: $u = \cos(x) + 4, \quad \frac{du}{dx} = -\sin(x).$

Then: $\int \frac{\sin(x)}{\cos(x) + 4} \, dx = -\int \frac{1}{u} \, du = -\ln|u| + C = -\ln|\cos(x) + 4| + C.$

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5. Find $\int x^2 (\ln x + 10) \, dx$.

Solution:

We split into two parts: $\int x^2 (\ln x + 10) \, dx = \int x^2 \ln x \, dx + \int 10x^2 \, dx.$

1. For $\int x^2 \ln x \, dx$, use integration by parts: Let $u = \ln x, \, dv = x^2 dx$. Then $du = \frac{1}{x} dx, \, v = \frac{x^3}{3}$. $\int x^2 \ln x \, dx = \frac{x^3}{3} \ln x - \int \frac{x^3}{3} \cdot \frac{1}{x} \, dx = \frac{x^3}{3} \ln x - \frac{x^3}{9}.$

2. For $\int 10x^2 \, dx$: $\int 10x^2 \, dx = \frac{10x^3}{3}.$

Combine the results: $\int x^2 (\ln x + 10) \, dx = \frac{x^3}{3} \ln x - \frac{x^3}{9} + \frac{10x^3}{3} + C.$

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Would you like detailed steps for any of these solutions or further clarifications? Here are 5 related questions you can explore:

1. How do you apply the product rule for derivatives in question 1?
2. Can we explore the chain rule for parametric differentiation in question 2?
3. How do we simplify higher-order derivatives, like in question 3?
4. What alternative substitutions could be used for integrals like in question 4?
5. How do you simplify results after applying integration by parts?

Tip: Always organize your work when solving multi-step derivatives or integrals to avoid errors!