Question - Derivative of a Composition of Functions using the Chain Rule
mathderivative calculationderivative composition functionschain rule differentiationfunction differentiation
Solution:
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.
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