Differentiate the function y = x^4.

The derivative of y = x^4 is 4x^3.

To differentiate y = x^4, we use the power rule of differentiation. This states that if y = x^n, then dy/dx = nx^(n-1). Applying this rule to y = x^4, we get:

dy/dx = 4x^3

Therefore, the derivative of y = x^4 is 4x^3. This means that the slope of the tangent line to the graph of y = x^4 at any point (x, y) is given by 4x^3.

We can also use this derivative to find the critical points of the function. These are the points where the derivative is equal to zero or undefined. In this case, the derivative is never undefined, but it is equal to zero when x = 0. Therefore, the critical point of y = x^4 is (0, 0).

Finally, we can use the derivative to determine the concavity of the graph of y = x^4. The second derivative of y = x^4 is given by:

d^2y/dx^2 = 12x^2

If d^2y/dx^2 > 0, then the graph is concave up. If d^2y/dx^2 < 0, then the graph is concave down. If d^2y/dx^2 = 0, then the graph has an inflection point. In this case, d^2y/dx^2 is always positive, so the graph of y = x^4 is concave up for all values of x.