Derivative of Square Root Function using Chain Rule: d/dx(sqrt(5x^2 + 4x + 1))

Published on March 3, 2026

Calculus

Calculate the derivative of y = sqrt(5x^2 + 4x + 1) by applying the Chain Rule, resulting in (5x + 2) / sqrt(5x^2 + 4x + 1).

Question

Answer

The user has presented an image asking for the derivative of the function:

We need to find

This requires the use of the chain rule.
First, rewrite the square root as a power:

Let

. Then

The chain rule states:

Step 1: Find
Using the power rule for differentiation (

Substituting back

Step 2: Find

Using the rules for differentiation of sums and powers:

Step 3: Apply the Chain Rule

Step 4: Simplify the result

We can factor out a 2 from the numerator:

Substitute this back:

Cancel the factor of 2:

The derivative is:

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