Differentiate the function y = sqrt(x).

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

To differentiate y = sqrt(x), we can use the power rule of differentiation. Let y = x^(1/2), then:

dy/dx = (1/2)x^(-1/2)

Simplifying this expression, we get:

dy/dx = 1/(2sqrt(x))

Therefore, the derivative of y = sqrt(x) is 1/(2sqrt(x)). This means that the slope of the tangent line to the curve y = sqrt(x) at any point (x, sqrt(x)) is given by 1/(2sqrt(x)).

To learn more about how to use the power rule and other basic rules for differentiation, you can review the basic differentiation rules.

We can also use this derivative to find the equation of the tangent line to the curve y = sqrt(x) at a specific point. For example, if we want to find the equation of the tangent line at x = 4, we first find the slope of the tangent line using the derivative:

dy/dx = 1/(2sqrt(x))
dy/dx at x = 4 = 1/(2sqrt(4)) = 1/4

So the slope of the tangent line at x = 4 is 1/4. To find the equation of the tangent line, we need a point on the line. Since the point (4, 2) lies on the curve y = sqrt(x), it also lies on the tangent line. Therefore, the equation of the tangent line is:

y - 2 = (1/4)(x - 4)

Simplifying this equation, we get:

y = (1/4)x + 1.5

So the equation of the tangent line to the curve y = sqrt(x) at x = 4 is y = (1/4)x + 1.5.

For further details on deriving and understanding the equations of tangent lines, check out this explanation on the equation of a tangent line.

If you are just beginning to study derivatives, I recommend starting with an introduction to derivatives which will provide foundational concepts and examples to help you understand how differentiation is applied across various functions.