If y=`(3x^2-9x+5)^2` then dy/dx=

Options

A. `2(3x^2-9x+5)(6x-5)`
B. `2(3x^2-9x+5)`
C. `3(3x^2-9x+5)`
D. `6(3x^2-9x+5)(2x-3)`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = (3x^2 - 9x + 5)^2\), we will use the chain rule of differentiation. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
Let \(u = 3x^2 - 9x + 5\). Then, we can rewrite \(y\) as:
\[ y = u^2 \]

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \(y\) with respect to \(u\) is:
\[ \frac{dy}{du} = 2u \]

### Step 3: Differentiate the inner function
Now, we need to differentiate \(u\) with respect to \(x\):
\[ \frac{du}{dx} = \frac{d}{dx}(3x^2 - 9x + 5) \]
Calculating this gives:
\[ \frac{du}{dx} = 6x - 9 \]

### Step 4: Apply the chain rule
Now we can apply the chain rule to find \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 2u \cdot (6x - 9) \]

### Step 5: Substitute back for \(u\)
Now, substitute \(u\) back into the equation:
\[ \frac{dy}{dx} = 2(3x^2 - 9x + 5)(6x - 9) \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 2(3x^2 - 9x + 5)(6x - 9) \]