Use implicit differentiation to find an equation of the tangent line to the curve 3xy^3+4xy=63 at the point (9,1)(9,1).
3xy^3+4xy=63
3y^3 + 9xy^2 y' + 4y + 4xy' = 0
y'(9xy^2+4x) = -(3y^3+4y)
y' = -(3y^3+4y)/(9xy^2+4x)
So, at (9,1) y' = -7/117
Now you have a point and a slope, so the line is
y-1 = -7/117 (x-9)
Refer to
http://www.wolframalpha.com/input/?i=plot+3xy%5E3%2B4xy%3D63,+y-1+%3D+-7%2F117+(x-9)+for+0%3C%3Dx%3C%3D20
To find the equation of the tangent line to the curve at the point (9,1), we will use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x, treating y as a function of x.
On the left side, we use the product rule: d/dx(x*y^3) = x*(d/dx(y^3)) + y^3*(d/dx(x))
On the right side, the derivative of a constant is zero.
So, the differentiation of the equation becomes:
3*x*d/dx(y^3) + 3*y^2*x + 4*d/dx(x*y) = 0
Step 2: Simplify the equation.
The derivative of y^3 with respect to x (d/dx(y^3)) is 3y^2*(dy/dx) by the chain rule.
The derivative of x*y with respect to x (d/dx(x*y)) is y + x*(dy/dx) by the product rule.
Substituting these derivatives in the equation, we get:
3*x*3y^2*(dy/dx) + 3*y^2*x + 4*(y + x*(dy/dx)) = 0
Step 3: Solve for dy/dx.
Rearranging the equation to isolate the term involving dy/dx, we have:
9*x*y^2*(dy/dx) + 4*x*(dy/dx) = -(3*y^2*x + 4*y)
Factoring out (dy/dx), we get:
(9*x*y^2 + 4*x)*(dy/dx) = -(3*y^2*x + 4*y)
Step 4: Find the value of dy/dx at the point (9,1).
Substitute x = 9 and y = 1 into the equation to evaluate it at the point (9,1):
(9*9*1^2 + 4*9)*(dy/dx) = -(3*(1^2)*9 + 4*1)
81 + 36*(dy/dx) = -(27 + 4)
81 + 36*(dy/dx) = -31
Step 5: Solve for dy/dx.
Subtract 81 from both sides:
36*(dy/dx) = -112
Divide both sides by 36:
dy/dx = -112 / 36
dy/dx = -14/9
Step 6: Write the equation of the tangent line.
Now that we have the slope of the tangent line (dy/dx = -14/9), we can use the point-slope form of the equation of a line to write the equation of the tangent line.
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the given point.
Substituting the values into the equation:
y - 1 = (-14/9)(x - 9)
Multiplying through by 9 to eliminate the fraction, we get:
9y - 9 = -14x + 126
Rearranging the equation in slope-intercept form, we have:
9y = -14x + 135
Dividing both sides by 9, we get:
y = (-14/9)x + 15
Therefore, the equation of the tangent line to the curve at the point (9,1) is y = (-14/9)x + 15.
To find the equation of the tangent line to the curve using implicit differentiation, we first differentiate both sides of the given equation with respect to x.
Let's differentiate the equation 3xy^3 + 4xy = 63 with respect to x.
The derivative of 3xy^3 with respect to x can be found using the product rule. The derivative with respect to x of y^3 is (3y^2)(dy/dx), and the derivative with respect to x of 3xy^3 is 3y^3 + 3xy^2 (dy/dx).
Similarly, the derivative of 4xy with respect to x can be found using the product rule. The derivative with respect to x of xy is (y)(dx/dx) + (x)(dy/dx), which simplifies to y + x(dy/dx).
Lastly, the derivative of 63 with respect to x is 0 since 63 is a constant.
Putting everything together, we get:
3y^3 + 3xy^2(dy/dx) + y + x(dy/dx) = 0
Now, let's find the slope of the tangent line by substituting the values of x=9 and y=1 into the differentiated equation.
Substituting x=9 and y=1, we get:
3(1)^3 + 3(9)(1)^2(dy/dx) + 1 + 9(dy/dx) = 0
Simplifying, we have:
3 + 27(dy/dx) + 1 + 9(dy/dx) = 0
Combine like terms:
36(dy/dx) + 4 = 0
To find the value of dy/dx, we isolate it:
36(dy/dx) = -4
dy/dx = -4/36 = -1/9
So, the slope of the tangent line at the point (9,1) is -1/9.
Now that we know the slope of the tangent line and a point on the line (9,1), we can use the point-slope form of a line to find the equation of the tangent line.
Using the equation y - y1 = m(x - x1), where (x1, y1) is the point (9,1) and m is the slope (-1/9), we have:
y - 1 = (-1/9)(x - 9)
Simplifying further, we get:
y - 1 = (-1/9)x + 1
y = (-1/9)x + 2
Therefore, the equation of the tangent line to the curve 3xy^3 + 4xy = 63 at the point (9,1) is y = (-1/9)x + 2.