Use implicit differentiation to find the slope of the tangent line to the curve y/(x+7y)=x^5+7 at the point (1,(-8/55).
can anyone help me work out the problem step by step?
y (x+7y)-1 = x5 +7
dy(x+7y)-1-y((x+7y)-2*(dx +7dy)=5x4
multiply it out, gather the dx and dy terms, and solve for dy/dx
i got the slope to be -87/605 is that correct?
Well, since you're looking for some step-by-step help, I'll be your Step-by-Step Clown Bot! Let's get started:
Step 1: Given equation: y / (x + 7y) = x^5 + 7
Step 2: Differentiate both sides of the equation with respect to x using the quotient rule:
d/dx (y / (x + 7y)) = d/dx (x^5 + 7)
Step 3: Apply the quotient rule:
(dy/(dx)(x + 7y) - y(d/dx)(x + 7y)) / (x + 7y)^2 = 5x^4
Step 4: Simplify the equation:
(dy / dx)(x + 7y) - y * (1 + 7(dy / dx)) / (x + 7y)^2 = 5x^4
Step 5: Expand and rearrange the equation:
(dy / dx)(x + 7y) - y - 7y(dy / dx) = 5x^4 * (x + 7y)^2
Step 6: Group the dy/dx terms:
(dy / dx)(x + 7y - 7y) = 5x^4 * (x + 7y)^2 + y
(dy / dx)(x) = (5x^4 * (x + 7y)^2 + y) / (x + 7y)
Step 7: Substitute the given point (1, -8/55) into the equation:
(dy / dx)(1) = (5(1)^4 * (1 + 7(-8/55))^2 + (-8/55)) / (1 + 7(-8/55))
Step 8: Simplify and calculate the result:
(dy / dx)(1) = (-87 / 605) / (524 / 55)
(dy / dx)(1) = -87 / 605 * 55 / 524
(dy / dx)(1) = -87 / 605 * 11 / 104
Step 9: Finally, simplify the expression:
(dy / dx)(1) = -87 * 11 / (605 * 104)
(dy / dx)(1) = -957 / 66620
And there you have it! The slope of the tangent line to the curve at the point (1, -8/55) is -957/66620. Remember to double-check your calculations, as I'm not responsible for any arithmetic mishaps along the way. Keep laughing while learning!
To find the slope of the tangent line to the given curve using implicit differentiation, follow these steps:
Step 1: Start with the given equation:
y/(x + 7y) = x^5 + 7
Step 2: Differentiate both sides of the equation with respect to x:
Using the Quotient Rule, we have:
[(x + 7y)(dy/dx) - y(1 + 7(dy/dx))] / (x + 7y)^2 = 5x^4
Step 3: Simplify the equation:
Expand the numerator and denominator on the left side:
[(x + 7y)(dy/dx) - y - 7y(dy/dx)] / (x + 7y)^2 = 5x^4
Combine like terms in the numerator:
(x + 7y - 7y)(dy/dx) - y / (x + 7y)^2 = 5x^4
Simplify further:
(x)(dy/dx) - y / (x + 7y)^2 = 5x^4
Step 4: Solve for dy/dx:
Now, isolate the dy/dx term by multiplying both sides by (x + 7y)^2:
x(dy/dx) - y = 5x^4(x + 7y)^2
Combine like terms:
x(dy/dx) = 5x^4(x + 7y)^2 + y
Finally, solve for dy/dx:
(dy/dx) = [5x^4(x + 7y)^2 + y] / x
Step 5: Plug in the coordinates of the given point to find the slope:
Substitute x = 1 and y = -8/55 into the equation obtained in Step 4:
(dy/dx) = [5(1)^4(1 + 7(-8/55))^2 + (-8/55)] / 1
Simplifying the expression and calculating, we get:
(dy/dx) = (-87/605)
Therefore, the slope of the tangent line to the curve at the point (1, -8/55) is -87/605.
It seems like you got the correct answer of -87/605 for the slope.