Find the equation of the tangent to the curve sin(x+y)=xy at the origin.
differentiate implicitly
cos(x+y) (1 + dy/dx) = x dy/dx + y
at (0,0_
cos(0)(1+dy/dx) = 0+0
1(1+dy/dx) = 0
dy/dx = -1
so slope is -1 , and y-intercept is 0
equation of tangent: y = -x
Thank you!!!
To find the equation of the tangent to the curve at the origin, we need to calculate the derivative of the curve and substitute the values of x and y with the coordinates of the origin.
Step 1: Calculate the derivative of the given curve.
To find the derivative of the curve sin(x+y) = xy, we will use implicit differentiation.
Differentiating both sides with respect to x:
cos(x+y) * (1 + dy/dx) = y + xy'
Step 2: Substitute the values of x and y with the coordinates of the origin.
Since we want to find the equation of the tangent at the origin, we need to substitute x = 0 and y = 0.
cos(0+0) * (1 + dy/dx) = 0 + 0 * y'
cos(0) * (1 + dy/dx) = 0
1 * (1 + dy/dx) = 0
1 + dy/dx = 0
dy/dx = -1
Step 3: Write the equation of the tangent.
Now that we have the derivative dy/dx = -1, we can write the equation of the tangent in point-slope form using the coordinates of the origin (0,0).
y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope.
Plugging in (0,0) as (x1, y1) and -1 as m:
y - 0 = -1(x - 0)
y = -x
Therefore, the equation of the tangent to the curve sin(x+y) = xy at the origin is y = -x.