If y = sqrt(4x^2 + 1), find dy/dx.

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Question: If y = sqrt(4x^2 + 1), find dy/dx.

Options:

Correct Answer: (4x)/(sqrt(4x^2 + 1))

Solution:

Using the chain rule, dy/dx = (4x)/(2sqrt(4x^2 + 1)) = (4x)/(sqrt(4x^2 + 1)).

If y = sqrt(4x^2 + 1), find dy/dx.

If y = sqrt(4x^2 + 1), find dy/dx.

Correct Answer: 4x / sqrt(4x^2 + 1)

Step 1: Identify the function y = sqrt(4x^2 + 1).
Step 2: Rewrite the square root as an exponent: y = (4x^2 + 1)^(1/2).
Step 3: Use the chain rule to differentiate. The chain rule states that if you have a function of a function, you multiply the derivative of the outer function by the derivative of the inner function.
Step 4: Differentiate the outer function (1/2)(4x^2 + 1)^(-1/2) and keep the inner function (4x^2 + 1) the same.
Step 5: Now differentiate the inner function 4x^2 + 1, which gives you 8x.
Step 6: Combine the results from Step 4 and Step 5: dy/dx = (1/2)(4x^2 + 1)^(-1/2) * 8x.
Step 7: Simplify the expression: dy/dx = (4x)/(sqrt(4x^2 + 1)).

Chain Rule – The chain rule is used to differentiate composite functions, allowing us to find the derivative of y with respect to x when y is defined in terms of another function of x.
Square Root Differentiation – Differentiating functions that involve square roots requires careful application of the chain rule and understanding of how to handle the derivative of the square root function.