Q.Find $\frac{dy}{dx}$ in the following: $(3x^2 - 9x + 5)^9$

The key idea is Implicit Differentiation — but here we have an explicit function of $x$, so we simply apply the chain rule.

Let $y = (3x^2 - 9x + 5)^9$.

Differentiate the outer power first:

$\frac{dy}{dx} = 9(3x^2 - 9x + 5)^8 \cdot \frac{d}{dx}(3x^2 - 9x + 5)$.

Now differentiate the inner polynomial:

$\frac{d}{dx}(3x^2 - 9x + 5) = 6x - 9$.

Multiply:

$\frac{dy}{dx} = 9(3x^2 - 9x + 5)^8 (6x - 9)$.

Factor the constant: $6x - 9 = 3(2x - 3)$, so

$\frac{dy}{dx} = 27(2x - 3)(3x^2 - 9x + 5)^8$.

The problem asks for $\frac{dy}{dx}$ when $y = (3x^2 - 9x + 5)^9$. This is a straightforward application of the chain rule — no implicit differentiation is actually needed here, since $y$ is already expressed explicitly in terms of $x$. But the deeper idea is the same: we differentiate the "outer function" (the power of 9) and multiply by the derivative of the "inner function" (the quadratic).

Let’s walk through it.

1. Identify the structure. We have $y = [u(x)]^9$, where $u(x) = 3x^2 - 9x + 5$. The chain rule says:

2. Differentiate the inner function. $u(x) = 3x^2 - 9x + 5$ is a polynomial. Differentiate term by term:

(Recall: derivative of $3x^2$ is $6x$, derivative of $-9x$ is $-9$, derivative of constant $5$ is $0$.)

3. Assemble the result. Substitute $u(x)$ and $u'(x)$ into the chain rule expression:

4. Simplify if desired. The factor $(6x - 9)$ can be factored as $3(2x - 3)$, giving:

This is a cleaner form, but the previous expression is also perfectly acceptable.

Method: The Chain Rule for a Composite Function

Whenever a function is "wrapped inside" another function — $f(g(x))$ — differentiate the outer function first (with respect to its own argument), then multiply by the derivative of the inner function.

Steps

Step 1: Identify the outer function and the inner function

Write $y=f(u)$ where $u=g(x)$ is everything "inside" the outermost operation.

Step 2: Differentiate the outer function with respect to $u$

Use the standard derivative rule for whatever the outer function is (log, power, trig, exponential, ...), keeping $u$ untouched.

Step 3: Differentiate the inner function $u$ with respect to $x$

Step 4: Multiply the two results

If the inner function is itself composite (a function inside a function inside a function), repeat the process — multiply in one more derivative for each layer.

Applying to this problem: for $y=(3x^2-9x+5)^9$, the outer function is $u^9$ with $u=3x^2-9x+5$; $\frac{dy}{du}=9u^8$ and $\frac{du}{dx}=6x-9$, so $\frac{dy}{dx}=9(3x^2-9x+5)^8(6x-9)$, which factors to $27(2x-3)(3x^2-9x+5)^8$.

Common Mistakes

Mistake 1: Forgetting to multiply by the derivative of the inner polynomial.

Why it's wrong: writing $9(3x^2-9x+5)^8$ alone (applying only the power rule to the outer power) leaves out the essential $(6x-9)$ factor from the inner function. Correct approach: always differentiate what's inside the parentheses as its own explicit step.

Mistake 2: Leaving the constant un-factored, risking a copying error in the final boxed answer.

Why it's wrong: $9(6x-9)$ can be simplified to $27(2x-3)$, and skipping this factoring step (while not incorrect) makes it easy to mis-transcribe the coefficient when writing the final answer. Correct approach: factor common constants out before finalising.