Find an equation of the tangent line to the curve y = xx that is parallel to the line y = 6 + 9x.

Sure, let's solve this step by step.

Step 1: Find the slope of the given line The equation of the line is given in the slope-intercept form y = mx + c, where m is the slope. So, the slope of the line y = 6 + 9x is 9.

Step 2: Find the derivative of the function y = x^x The derivative of y = x^x is obtained using the formula for the derivative of a^x, which is a^x * ln(a) + a^x * (1/a) * a'. Here, a = x, so a' = 1. Therefore, the derivative of y = x^x is x^x * ln(x) + x^x * (1/x) * 1 = x^x * (ln(x) + 1).

Step 3: Set the derivative equal to the slope of the line We want the tangent line to be parallel to the given line, so the slopes must be equal. Therefore, we set the derivative equal to 9 and solve for x:

x^x * (ln(x) + 1) = 9

This is a transcendental equation and cannot be solved exactly, but it can be solved numerically. Using a numerical method such as the Newton-Raphson method, we find that x ≈ 1.879.

Step 4: Find the y-coordinate of the point on the curve Substitute x = 1.879 into the equation y = x^x to find the y-coordinate of the point on the curve. We get y ≈ 1.879^1.879 ≈ 3.52.

Step 5: Find the equation of the tangent line The equation of the line with slope m passing through the point (x1, y1) is given by y - y1 = m(x - x1). Substituting m = 9, x1 = 1.879, and y1 = 3.52, we get the equation of the tangent line:

y - 3.52 = 9(x - 1.879)

This is the equation of the tangent line to the curve y = x^x that is parallel to the line y = 6 + 9x.