The derivative of 𝑦 = √𝑥2+33𝑥

To find the derivative of the function y = √(x^2 + 33x), we will use the chain rule. The chain rule is a formula to compute the derivative of a composite function. Here's the step-by-step solution:

1. Rewrite the function: y = (x^2 + 33x)^(1/2)

2. Apply the chain rule: dy/dx = (1/2)(x^2 + 33x)^(-1/2) * derivative of (x^2 + 33x)

3. Find the derivative of (x^2 + 33x), which is 2x + 33.

4. Substitute this into the equation from step 2: dy/dx = (1/2)(x^2 + 33x)^(-1/2) * (2x + 33)

5. Simplify the equation to get the final derivative: dy/dx = (2x + 33) / (2√(x^2 + 33x))

So, the derivative of y = √(x^2 + 33x) is dy/dx = (2x + 33) / (2√(x^2 + 33x)).