# How to differentiate square root function √(1 + x + x^{2} + 2x^{3})?

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 + x^{2} + 2x^{3}) is (1 + 2x + 6x^{2}) / 2√(1 + x + x^{2} + 2x^{3}).

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 + x^{2} + 2x^{3})] = [ 1 / 2√(1 + x + x^{2} + 2x^{3}) ] × d/dx (1 + x + x^{2} + 2x^{3}).

⇒ Finally, d/dx [√(1 + x + x^{2} + 2x^{3})] = [ 1 / 2√(1 + x + x^{2} + 2x^{3}) ] × (1 + 2x + 6x^{2}).