### Lecture on the Chain Rule for Derivatives

Today, we'll dive into the chain rule for derivatives, a fundamental concept in calculus that allows us to find the derivative of a composite function. To provide a clear example, let's examine the derivative of the composite function $(x^3)^2$. We'll tackle this problem in two ways: first, by simplifying the expression and then using the power rule, and second, by directly applying the chain rule.

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#### Simplifying Before Differentiation

First, let's simplify $(x^3)^2$. According to the laws of exponents, when we raise a power to another power, we multiply the exponents:

$$
(x^3)^2 = x^{3\cdot2} = x^6
$$

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Now, to find the derivative of $x^6$, we apply the power rule. The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Applying this to $x^6$, we obtain:

$$
\frac{d}{dx}x^6 = 6x^{6-1} = 6x^5
$$

By simplifying the expression first and then differentiating, we find that the derivative of $(x^3)^2$ is $6x^5$.

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#### Using the Chain Rule

$\dfrac{dy}{dx}=\dfrac{dy}{du} \cdot \dfrac{du}{dx}$

to find the derivative of $y=\left(x^3\right)^2$

Now, let's approach the same problem using the chain rule. The chain rule states that if we have a composite function $f(u)$, then the derivative is given by $f'(u) \cdot u'$. In our example, let $u = x^3$ be the inner function and $y = u^2$ be the outer function.

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First, we find the derivative of the outer function with respect to $u$, treating $u$ as the variable. If $y = u^2$, then:

$$
\frac{dy}{du} = 2u^{2-1} = 2u
$$

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Next, we find the derivative of the inner function $u = x^3$ with respect to $x$:

$$
\frac{du}{dx} = 3x^{3-1} = 3x^2
$$

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Applying the chain rule:

$$
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (2u) \cdot (3x^2)
$$

Substituting $u = x^3$ back into our derivative gives us:

$$
\frac{dy}{dx} = 2(x^3) \cdot 3x^2 = 6x^5
$$

Interestingly, both methods yield the same result: $6x^5$

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This not only confirms the reliability of calculus principles but also demonstrates two different methodologies for solving the same problem.

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#### Conclusion

This comparison illustrates the versatility in selecting a differentiation technique based on the specific problem. Simplifying before differentiating can often streamline the process, making it more straightforward. However, the chain rule proves indispensable for dealing with composite functions directly, especially when simplification is not straightforward. This underscores the importance and utility of calculus tools in mathematical analysis and problem-solving.