Differentiate the function y = x^5.

The derivative of y = x^5 is 5x^4.

To differentiate y = x^5, 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^5, we get:

dy/dx = 5x^(5-1)
dy/dx = 5x^4

Therefore, the derivative of y = x^5 is 5x^4. This means that the slope of the tangent line to the graph of y = x^5 at any point (x, y) is equal to 5x^4. We can use this derivative 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^5 at x = 2, we first find the derivative at x = 2:

dy/dx = 5x^4
dy/dx = 5(2)^4
dy/dx = 80

This means that the slope of the tangent line at x = 2 is 80. To find the equation of the tangent line, we also need a point on the line. We can use the point (2, 32), which is on the graph of y = x^5 at x = 2. Using the point-slope form of a line, we get:

y - 32 = 80(x - 2)
y - 32 = 80x - 160
y = 80x - 128

Therefore, the equation of the tangent line to the graph of y = x^5 at x = 2 is y = 80x - 128.