Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin(8x) = x cos(2y), (/2, /4)

Answer 1

To find the equation of the tangent line at the given point (π/2, π/4), implicit differentiation is used to find the slope of the curve, and then the point-slope form of the line equation is used with the calculated slope and the point to get the equation of the tangent line.

Step-by-step explanation:

To find the equation of the tangent line to the curve y sin(8x) = x cos(2y) at the given point (π/2, π/4), we first need to use implicit differentiation to find the derivative of the curve, which will give us the slope of the tangent line at the point. Differentiating both sides of the equation with respect to x, we get:

1. cos(8x) * y + sin(8x) * dy/dx = cos(2y) - 2x * sin(2y) * dy/dx
2. We need to solve this equation for dy/dx, which represents the slope of the tangent line.

Now substitute the coordinates of the given point into the differentiated equation to find the actual slope value. After calculating the slope, we then use the point-slope form of the equation of a line to write the equation of the tangent line:

• y - π/4 = slope * (x - π/2)

This gives us the final equation for the tangent line at the given point.