To solve the task, we need to analyze the problem related to differentiation rules for composite functions.

Summary of Key Information in the Image:

1. Theorem: It describes the chain rule for differentiating composite functions $y(u(x))$.

   • For $y'(u(x))$, the derivative is expressed as: $y'(u(x)) = y'_u \cdot u'_x,$ where:
     • $y'_u = \frac{dy}{du}$ is the derivative of the outer function $y(u)$ with respect to $u$,
     • $u'_x = \frac{du}{dx}$ is the derivative of the inner function $u(x)$ with respect to $x$.

2. Generalization to Multiple Compositions: For three nested functions $z = y(u(v(x)))$, the chain rule extends as: $\frac{d}{dx} y(u(v(x))) = y'_u \cdot u'_v \cdot v'_x.$

3. Task: Choose the correct equalities for the derivatives provided at the end of the text.

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Let's solve each of the options:

1. $(\cos(\sin(x)))' = -\sin(\cos(x))$

Using the chain rule for the composite function $\cos(\sin(x))$:

• Outer function: $y = \cos(u)$, derivative $y'_u = -\sin(u)$,
• Inner function: $u = \sin(x)$, derivative $u'_x = \cos(x)$,
• Combining: $\frac{d}{dx} \cos(\sin(x)) = -\sin(\sin(x)) \cdot \cos(x).$ This does not match $-\sin(\cos(x))$. Hence, this option is incorrect.

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2. $(\cos^2(x))' = 2\cos(x) \cdot (-\sin(x))$

For $\cos^2(x)$:

• Rewrite as $y = (\cos(x))^2$,
• Outer function: $y = u^2$, derivative $y'_u = 2u$,
• Inner function: $u = \cos(x)$, derivative $u'_x = -\sin(x)$,
• Combining: $\frac{d}{dx} \cos^2(x) = 2\cos(x) \cdot (-\sin(x)) = -2\cos(x)\sin(x).$ This matches the given expression. Hence, this option is correct.

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3. $(\cos^2(x))' = (\cos(x))' \cdot (\sin(x))'$

The expression suggests combining derivatives $(\cos(x))'$ and $(\sin(x))'$:

• $(\cos(x))' = -\sin(x)$,
• $(\sin(x))' = \cos(x)$,
• Combining these: $(\cos(x))' \cdot (\sin(x))' = (-\sin(x)) \cdot \cos(x) = -\sin(x)\cos(x).$ This is not the correct way to differentiate $\cos^2(x)$. Hence, this option is incorrect.

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4. $(\cos^2(x))' = 2\sin(x)$

From the earlier computation: $\frac{d}{dx} \cos^2(x) = -2\cos(x)\sin(x).$ This does not simplify to $2\sin(x)$. Hence, this option is incorrect.

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Final Answer:

The correct equality is:

• Option 2: $(\cos^2(x))' = 2\cos(x) \cdot (-\sin(x))$.

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Would you like me to clarify any part of this solution or go deeper into the chain rule? Let me know!

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5 Related Questions:

1. How does the chain rule apply to functions of multiple nested compositions?
2. Can the chain rule be extended to functions with more than three nested levels?
3. What are the common mistakes when applying the chain rule in derivatives?
4. How does the chain rule differ when applied to trigonometric versus polynomial functions?
5. Why is the derivative of $\cos^2(x)$ negative?

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Tip:

When applying the chain rule, always identify the outermost function first and work step by step inward to avoid errors!