Derivative of a Composition of Functions using the Chain Rule
The given function is \( y = 3(x^2 - x)^3 \). To find \( \frac{dy}{dx} \), we need to differentiate this function with respect to \( x \).
This is a composition of functions, and therefore we will need to apply the chain rule to differentiate it. The chain rule states that if we have a function \( h(x) = f(g(x)) \), then its derivative is \( h'(x) = f'(g(x)) \cdot g'(x) \).
In your case, let \( u = x^2 - x \). Then the function \( y \) can be rewritten as \( y = 3u^3 \).
Now, we will differentiate both sides with respect to \( x \):
1. Differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = 3 \cdot 3u^2 = 9u^2 \]
2. Differentiate \( u = x^2 - x \) with respect to \( x \):
\[ \frac{du}{dx} = 2x - 1 \]
Now, apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 9u^2(2x - 1) \]
Substitute \( u \) back in terms of \( x \):
\[ \frac{dy}{dx} = 9(x^2 - x)^2(2x - 1) \]
Remember to expand and simplify the expression if needed. But as per the instruction given in the question, you wanted the expression for \( \frac{dy}{dx} \), which is what we've found.
