Differentiate the following w.r.t.x :
`sqrt(15x^(2)-x+1)`

Answer

To differentiate the function \( y = \sqrt{15x^2 - x + 1} \) with respect to \( x \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
Let \( u = 15x^2 - x + 1 \). Then, we can rewrite \( y \) as:
\[ y = \sqrt{u} = u^{1/2} \]

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( y \) with respect to \( u \) is:
\[ \frac{dy}{du} = \frac{1}{2} u^{-1/2} = \frac{1}{2\sqrt{u}} \]

### Step 3: Differentiate the inner function
Now, we differentiate \( u \) with respect to \( x \):
\[ u = 15x^2 - x + 1 \]
\[ \frac{du}{dx} = \frac{d}{dx}(15x^2) - \frac{d}{dx}(x) + \frac{d}{dx}(1) = 30x - 1 + 0 = 30x - 1 \]

### Step 4: Apply the chain rule
Now we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{u}} \cdot (30x - 1) \]

### Step 5: Substitute back for \( u \)
Now we substitute back \( u = 15x^2 - x + 1 \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{15x^2 - x + 1}} \cdot (30x - 1) \]

### Final Answer
Thus, the derivative of \( y = \sqrt{15x^2 - x + 1} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{30x - 1}{2\sqrt{15x^2 - x + 1}} \]

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