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Differentiate the function y = sqrt(2x).

The derivative of y = sqrt(2x) is y' = 1/sqrt(2x).

To differentiate y = sqrt(2x), we can use the power rule of differentiation. This states that if y = x^n, then y' = nx^(n-1). In this case, n = 1/2, so we have:

y = (2x)^(1/2)
y' = (1/2)(2x)^(-1/2) * 2
y' = 1/sqrt(2x)

Therefore, the derivative of y = sqrt(2x) is y' = 1/sqrt(2x). This tells us the rate of change of y with respect to x at any point on the curve. For example, if x = 4, then y = sqrt(8) = 2sqrt(2), and y' = 1/sqrt(8) = sqrt(2)/4. This means that for every unit increase in x, y increases by sqrt(2)/4 units.

It is worth noting that the domain of y = sqrt(2x) is x >= 0, since the square root of a negative number is not defined in the real numbers. Therefore, the derivative is only defined for x > 0.

For a more detailed understanding of derivatives, consider reviewing introduction to derivatives, which explains fundamental concepts and techniques used in differentiation.

A-Level Maths Tutor Summary: To find the rate of change of y = sqrt(2x), we use the power rule for differentiation. By applying this rule, we learn that the derivative, y', is 1/sqrt(2x), showing how y changes with x. This formula is applicable when x > 0, since y is not defined for negative x. Understanding this helps us predict how y behaves as x increases. To explore various rules of differentiation in depth, you might find the page on differentiation rules helpful.

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