Use implicit differentiation to find the equation of the tangent line to the curve xy^3+xy=5 at the point (4,1) . The equation of this tangent line can be written in the form y = mx+b where m is:?
and where b is:?
ok i got result for m. but i don't know where is b.. i forget how do y=mx+b
see other post.
hey mily how u get result for m?
To find the equation of the tangent line to the curve at the point (4, 1), we can use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
Differentiating the equation xy^3 + xy = 5 with respect to x, we have:
d/dx(xy^3) + d/dx(xy) = d/dx(5)
Step 2: Apply the product rule and chain rule as needed.
Using the product rule, we find:
yd/dx(y^3) + 3y^2 + yd/dx(x) + x(dy/dx) = 0
Since we are looking for dy/dx (the slope of the tangent line), we can solve the equation above for dy/dx.
Step 3: Solve for dy/dx.
Rearranging the terms, we have:
x(dy/dx) + y + 3y^2(dy/dx) + y^3 = 0
Grouping the terms with dy/dx together, we get:
[dy/dx(x + 3y^2)] + [x + y^3] = 0
Dividing both sides by (x + 3y^2), we find:
dy/dx = -(x + y^3) / (x + 3y^2)
Now we have the derivative of y with respect to x, which gives us the slope of the tangent line at any point (x, y) on the curve.
Step 4: Find the slope at the point (4, 1).
Substituting x = 4 and y = 1 into the equation, we have:
m = -(4 + 1^3) / (4 + 3(1)^2)
m = -5 / 7
Therefore, the slope of the tangent line is m = -5/7.
Step 5: Find the y-intercept.
To find the y-intercept (b) of the tangent line, we can substitute the point (4, 1) and the slope (-5/7) into the equation y = mx + b and solve for b.
Substituting x = 4, y = 1, and m = -5/7, we have:
1 = (-5/7)(4) + b
1 = -20/7 + b
Multiplying both sides by 7 to eliminate the fraction, we get:
7 = -20 + 7b
Rearranging the equation:
7b = 20 + 7
7b = 27
Dividing both sides by 7:
b = 27/7
Therefore, the y-intercept of the tangent line is b = 27/7.
Hence, the equation of the tangent line to the curve xy^3 + xy = 5 at the point (4, 1) can be written as y = (-5/7)x + 27/7, where m = -5/7 and b = 27/7.