Problem

Find the slope of the tangent line to the curve
5sin(x)+4cos(y)3sin(x)cos(y)+x=7π
at the point (7π,7π/2).

Answer

Expert–verified
Hide Steps
Answer

1 is the slope of the tangent line to the curve at the point (7π,7π2)

Steps

Step 1 :Given the curve equation: x3sin(x)cos(y)+5sin(x)+4cos(y)7π=0

Step 2 :Differentiate with respect to x: x=3cos(x)cos(y)+5cos(x)+1

Step 3 :Differentiate with respect to y: y=3sin(x)sin(y)4sin(y)

Step 4 :Find the derivative of y with respect to x: dydx=3sin(x)sin(y)4sin(y)3cos(x)cos(y)+5cos(x)+1

Step 5 :Substitute the given point (7π,7π2) into the equation: dydx=3sin(7π)sin(7π2)4sin(7π2)3cos(7π)cos(7π2)+5cos(7π)+1

Step 6 :Simplify the expression: dydx=1

Step 7 :1 is the slope of the tangent line to the curve at the point (7π,7π2)

link_gpt