Use implicit differentiation to find an equation of the tangent line to the curve.
sin(x+y)=4x−4y at the point (π,π).
Tangent Line Equation:
sin(x+y)=4x−4y
cos(x+y)(1 + dy/dx) = 4 - 4dy/dx
cos(x+y) + (dy/dx)cos(x+y) = 4 - 4dy/dx
(dy/dx)[cos(x+y) + 4] = 4 - cos(x+y)
dy/dx = (4 - cos(x+y))/(cos(x+y) + 4)
at (π,π)
dy/dx = (4 - cos(2π))/(cos(2π) + 4)
= (4-1)/(1 + 4) = 3/5
y - π = (3/5)(x - π)
change to whichever variation you need
Oh, tangents...they're like lines that only get "acquainted" with curves at one point. Let's find out what this one likes to chat about! We'll use implicit differentiation to get the slope of the tangent at the point (π,π).
But first, let's remember one helpful rule of implicit differentiation: when you take the derivative of y with respect to x, you must chain it up!
So, let's start differentiating the given equation with respect to x, and keep in mind that y is actually a function of x:
cos(x+y) * (1 + dy/dx) = 4 - 4(dy/dx)
Now, let's plug in the values for x and y as (π,π):
cos(π+π) * (1 + dy/dx) = 4 - 4(dy/dx)
cos(2π) * (1 + dy/dx) = 4 - 4(dy/dx)
Since cos(2π) = 1, we can simplify the equation to:
1 * (1 + dy/dx) = 4 - 4(dy/dx)
Now let's solve for dy/dx:
1 + dy/dx = 4 - 4(dy/dx)
5dy/dx = 3
dy/dx = 3/5
So, the slope of the tangent line at the point (π,π) is 3/5.
Now, we can use the point-slope form of a line to find the equation of the tangent line at (π,π):
y - π = (3/5)(x - π)
And voila! The tangent line equation is:
y = (3/5)x + (2π/5) - π
or simplifying a bit:
y = (3/5)x + π/5
There you have it! The tangent line equation for the curve sin(x+y) = 4x - 4y at the point (π,π).
To find the equation of the tangent line, we need to first find the slope of the tangent line at the point (π, π) using implicit differentiation.
Differentiating both sides of the equation with respect to x, we get:
cos(x+y) * (1 + dy/dx) = 4 - 4(dy/dx)
Next, we need to solve for dy/dx, so we can find the slope of the tangent line. Rearranging the equation, we have:
cos(x+y) + 4(dy/dx) = 4 - cos(x+y)
Simplifying further, we get:
4(dy/dx) = 4 - cos(x+y) - cos(x+y)
4(dy/dx) = 4 - 2cos(x+y)
Now, let's substitute x = π and y = π into the equation. We have:
4(dy/dx) = 4 - 2cos(π+π)
4(dy/dx) = 4 - 2cos(2π)
Since cosine of 2π is equal to 1, we can simplify further:
4(dy/dx) = 4 - 2(1)
4(dy/dx) = 4 - 2
4(dy/dx) = 2
Finally, we can solve for dy/dx:
(dy/dx) = 2/4
(dy/dx) = 1/2
So, the slope of the tangent line at the point (π, π) is 1/2.
Now, using the point-slope form of a line, we can find the equation of the tangent line. We have:
y - y1 = m(x - x1)
Substituting values, we have:
y - π = (1/2)(x - π)
Simplifying, we get:
y - π = (1/2)x - π/2
y = (1/2)x - π/2 + π
y = (1/2)x + π/2
Therefore, the equation of the tangent line to the curve sin(x+y) = 4x - 4y at the point (π, π) is y = (1/2)x + π/2.
To find the equation of the tangent line to the curve, we first need to find the derivative of the given equation using implicit differentiation.
Step 1: Take the derivative of both sides of the equation with respect to x.
As the equation involves both x and y, we treat y as a function of x and apply the chain rule.
d/dx[sin(x+y)] = d/dx[4x - 4y]
Step 2: Apply the chain rule on the left side of the equation.
The derivative of sin(x+y) with respect to x can be expressed as:
cos(x+y) * (1 + dy/dx)
The term (1 + dy/dx) accounts for the derivative of the inner function y with respect to x.
Step 3: Derive the right side of the equation.
The derivative of 4x with respect to x is 4, and the derivative of -4y with respect to x is -4(dy/dx).
So, the equation becomes:
cos(x+y) * (1 + dy/dx) = 4 - 4(dy/dx)
Step 4: Solve for dy/dx.
Rearrange the equation to isolate dy/dx:
cos(x+y) + 4(dy/dx) * cos(x+y) = 4 - 4(dy/dx)
Simplifying the equation:
(cos(x+y) - 4 * cos(x+y)) * dy/dx = 4 - cos(x+y)
Dividing both sides by cos(x+y) - 4 * cos(x+y):
dy/dx = (4 - cos(x+y)) / (1 - 4)
dy/dx = (4 - cos(x+y)) / (-3)
Step 5: Substitute the coordinates of the given point into the derived expression to find dy/dx at that point.
The point given is (π,π), so we substitute x=π and y=π into the equation:
dy/dx = (4 - cos(π+π)) / (-3)
= (4 - cos(2π)) / (-3)
= (4 - 1) / (-3)
= -1
Hence, the derivative dy/dx is -1 at the point (π,π).
Step 6: Use point-slope form to find the tangent line equation.
The point-slope form of a line is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
Using the point (π, π) and the slope dy/dx = -1, we have:
y - π = -1(x - π)
Simplifying:
y - π = -x + π
Rearranging and simplifying further, we get the equation of the tangent line:
y = -x + 2π
Therefore, the equation of the tangent line to the curve sin(x+y) = 4x - 4y at the point (π, π) is y = -x + 2π.