Differentiate the function y = x^3.

The derivative of y = x^3 is 3x^2.

To differentiate y = x^3, we use the power rule of differentiation. The power rule states that if y = x^n, then dy/dx = nx^(n-1). Applying this rule to y = x^3, we get:

dy/dx = 3x^(3-1)
dy/dx = 3x^2

Therefore, the derivative of y = x^3 is 3x^2. This means that the slope of the tangent line to the graph of y = x^3 at any point (x, y) is equal to 3x^2. We can use this information to find the equation of the tangent line at a specific point.

For example, if we want to find the equation of the tangent line to the graph of y = x^3 at the point (2, 8), we first find the slope of the tangent line using the derivative:

dy/dx = 3x^2
dy/dx at x = 2 = 3(2)^2 = 12

So the slope of the tangent line at x = 2 is 12. To find the equation of the tangent line, we use the point-slope form:

y - y1 = m(x - x1)
y - 8 = 12(x - 2)
y - 8 = 12x - 24
y = 12x - 16

Therefore, the equation of the tangent line to the graph of y = x^3 at the point (2, 8) is y = 12x - 16.