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.