How to differentiate square root function √(1 + x + x² + 2x³)?

Differentiation or derivatives have many applications in various fields like engineering and science. They are used to find tangents and normals to various curves at different points.

Answer: The derivative of the square root function √(1 + x + x² + 2x³) is (1 + 2x + 6x²) / 2√(1 + x + x² + 2x³).

Let's understand the solution in detail.

Explanation:

We use the chain to differentiate a square root function like the one which is given.

Now, we follow the below steps to solve the question:

⇒ First, we differentiate the square root using the formula of d/dx (x)ⁿ = n(x)^{n - 1}. Now, if n = 1/2 (for square root), then d/dx (x)^1/2 = 1/2 × (x)^{1/2 - 1} = 1 / (2√x).

Next, we use the chain rule:

⇒ The expression becomes d/dx [√(1 + x + x² + 2x³)] = [ 1 / 2√(1 + x + x² + 2x³) ] × d/dx (1 + x + x² + 2x³).

⇒ Finally, d/dx [√(1 + x + x² + 2x³)] = [ 1 / 2√(1 + x + x² + 2x³) ] × (1 + 2x + 6x²).

Hence, the derivative of the square root function √(1 + x + x² + 2x³) is (1 + 2x + 6x²) / 2√(1 + x + x² + 2x³).