`sqrt(ax^2+bx+c)`

Answer

To differentiate the function \( f(x) = \sqrt{ax^2 + bx + c} \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Rewrite the function
We start by rewriting the function in a form that is easier to differentiate:
\[ f(x) = (ax^2 + bx + c)^{1/2} \]

**Hint:** Remember that the square root can be expressed as a power of \( \frac{1}{2} \).

### Step 2: Apply the chain rule
Using the chain rule, we differentiate \( f(x) \):
\[ f'(x) = \frac{1}{2} (ax^2 + bx + c)^{-1/2} \cdot \frac{d}{dx}(ax^2 + bx + c) \]

**Hint:** The chain rule states that the derivative of \( g(h(x)) \) is \( g'(h(x)) \cdot h'(x) \).

### Step 3: Differentiate the inner function
Now, we need to differentiate the inner function \( ax^2 + bx + c \):
\[ \frac{d}{dx}(ax^2 + bx + c) = 2ax + b \]

**Hint:** Remember that the derivative of a constant is zero, and use the power rule for \( x^2 \).

### Step 4: Substitute back into the derivative
Now we substitute back into our expression for \( f'(x) \):
\[ f'(x) = \frac{1}{2} (ax^2 + bx + c)^{-1/2} \cdot (2ax + b) \]

**Hint:** Keep track of the order of operations when substituting back.

### Step 5: Simplify the expression
Finally, we can simplify the expression:
\[ f'(x) = \frac{2ax + b}{2\sqrt{ax^2 + bx + c}} \]

**Hint:** Simplifying fractions can often make the final answer clearer.

### Final Answer:
Thus, the derivative of \( f(x) = \sqrt{ax^2 + bx + c} \) is:
\[ f'(x) = \frac{2ax + b}{2\sqrt{ax^2 + bx + c}} \]