Use implicit differentiation to find an equation of the tangent line to the curve
sin(x+y)=2x−2y at the point (π,π).
Tangent Line Equation:
To find the equation of the tangent line to the curve sin(x+y) = 2x - 2y at the point (π, π), we first need to find the derivative of the curve with respect to x.
Implicit differentiation:
Differentiating both sides with respect to x using chain rule, we have:
cos(x+y)(1+dy/dx) = 2 - 2(dy/dx) ⟹ cos(x+y)(1+dy/dx) - 2 + 2(dy/dx) = 0
Now we need to solve for dy/dx:
(dy/dx)(2+cos(x+y)) = 2 - cos(x+y) ⟹ dy/dx = (2 - cos(x+y)) / (2 + cos(x+y))
Now we calculate dy/dx at the point (π, π):
dy/dx = (2 - cos(2π)) / (2 + cos(2π)) = (2 - 1) / (2+1) = 1/3.
Now we have the slope (dy/dx) of the tangent line at the point (π, π).
To find the equation of the tangent line, we use the point-slope form:
y - y1 = m(x - x1)
Where m = dy/dx, x1 = π and y1 = π. Plugging in the values, we get:
y - π = 1/3 (x - π)
Now, we can rewrite the equation of the tangent line to a more standard form:
y = (1/3)x - (1/3)π + π
So, the equation of the tangent line is:
y = (1/3)x + (2/3)π.
To find the equation of the tangent line to the curve sin(x+y) = 2x - 2y, we will use implicit differentiation. Implicitly differentiating both sides of the equation with respect to x will allow us to find the derivative dy/dx.
Let's start by differentiating both sides of the equation with respect to x:
d/dx(sin(x+y)) = d/dx(2x-2y)
Using the chain rule on the left side, where sin(u) has derivative cos(u) * du/dx:
cos(x+y) * (d/dx(x+y)) = 2
Since we are looking for the slope of the tangent line at the point (π,π), we substitute x = π and y = π into the equation:
cos(π+π) * (d/dx(π+π)) = 2
Simplifying:
cos(2π) * (d/dx(2π)) = 2
cos(2π) is equal to 1, and the derivative of a constant (d/dx(2π)) is 0, so the equation becomes:
1 * 0 = 2
0 = 2
Since this equation is not true, it means that there is no well-defined derivative at the point (π,π). Therefore, the curve does not have a well-defined tangent line at that point.
As a result, the equation of the tangent line at the point (π,π) does not exist.
To find the equation of the tangent line to the curve, we will use implicit differentiation. Follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
Using the chain rule, we differentiate sin(x + y) with respect to x as follows:
d/dx[sin(x + y)] = d/dx[2x - 2y]
The derivative of sin(x + y) with respect to x is cos(x + y) * (1 + dy/dx) by the chain rule.
The derivative of 2x - 2y with respect to x is 2.
The equation after differentiating becomes:
cos(x + y) * (1 + dy/dx) = 2
Step 2: Solve the equation for dy/dx.
To find dy/dx, move cos(x + y) to the denominator and isolate dy/dx by dividing both sides of the equation by cos(x + y):
[(1 + dy/dx) / cos(x + y)] = 2
Simplifying further:
1 + dy/dx = 2 * cos(x + y)
Step 3: Substitute the given point into the equation.
Given point: (π, π)
Substitute x = π and y = π into the equation:
1 + dy/dx = 2 * cos(π + π)
1 + dy/dx = 2 * cos(2π)
Since cos(2π) = cos(0) = 1, the equation becomes:
1 + dy/dx = 2
Step 4: Solve for dy/dx.
Subtract 1 from both sides of the equation:
dy/dx = 2 - 1
dy/dx = 1
Step 5: Obtain the equation of the tangent line using the point-slope form.
Using the point (π, π) and the slope dy/dx = 1, the equation of the tangent line can be obtained. The point-slope form of a line is:
y - y₁ = m(x - x₁)
Plugging in the values:
y - π = 1(x - π)
y - π = x - π
y = x
Therefore, the equation of the tangent line to the curve sin(x + y) = 2x - 2y at the point (π, π) is y = x.