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 differentiate 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 \( x \) is given by:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
First, we differentiate the outer function \( y = u^2 \):
\[ \frac{dy}{du} = 2u \]

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

### Step 4: Combine the derivatives
Now we can substitute \( u \) back into our derivative:
\[ \frac{dy}{dx} = 2u \cdot \frac{du}{dx} = 2(3x^2 - 9x + 5)(6x - 9) \]

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

### Summary
The final answer is:
\[ \frac{dy}{dx} = 2(3x^2 - 9x + 5)(6x - 9) \]

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