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.
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).