Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin(12x) = x cos(2y), (/2, /4)

Answer 1

The equation of the tangent line at the point (π/2, π/4) on the curve y sin(12x) = x cos(2y) is found using implicit differentiation to determine the slope m, followed by applying the point-slope form y = mx + b.

Step-by-step explanation:

To find the equation of the tangent line to the curve y sin(12x) = x cos(2y) at the point (π/2, π/4), implicit differentiation is used. Differentiating both sides of the equation with respect to x gives:

y' sin(12x) + y cos(12x) * 12 = cos(2y) - x sin(2y) * 2y'

At the point (π/2, π/4), we plug in the x and y values to solve for the derivative y', which is the slope of the tangent line. After finding the slope, we use the point-slope form to write the equation of the tangent line. The final result for the slope and the tangent line equation will be:
Tangent Line Equation: y = mx + b, where m is the slope and b is the y-intercept.