Final answer:
To find the equation of the tangent line to the curve at the point (1,4), one must differentiate the equation implicitly to get the slope and then use the point-slope form to get the equation of the line, resulting in the equation y = 9x - 5.
Step-by-step explanation:
To find the equation of the tangent line to the curve y4 = 256x9 at the point (1,4), we first need to find the slope of the tangent line at that point. This involves taking the derivative of the given equation with respect to x, using the chain rule to differentiate implicitly.
First, differentiate each side of the equation: 4y3dy/dx = 256 × 9x8.
At the point (1,4), we substitute y = 4 into the derivative to get the slope: 4(4)3dy/dx = 256 × 9(1)8, which simplifies to dy/dx = 9.
Using the point-slope form of the linear equation, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point, we plug in the slope and the coordinates of the point to obtain the equation of the tangent line: y - 4 = 9(x - 1).
Finally, we simplify the equation to get the tangent line in slope-intercept form, which results in y = 9x - 5.