use implicit differentiation to find the slope of the tangent line to the curve (y)/(x-7y)=x^6+5 at the point (1,6/43)
To find the slope of the tangent line to the curve using implicit differentiation, follow these steps:
Step 1: Start with the given equation:
(y) / (x - 7y) = x^6 + 5
Step 2: Differentiate both sides of the equation with respect to x, using the chain rule where necessary. Treat y as a function of x:
d/dx [(y) / (x - 7y)] = d/dx (x^6 + 5)
Step 3: Apply the quotient rule on the left side:
[(x - 7y) * (d/dx)(y) - (y) * (d/dx)(x - 7y)] / (x - 7y)^2 = 6x^5
Step 4: Simplify the equation by finding the derivatives:
(x - 7y) * (dy/dx) - y * (1 - 7(dy/dx)) = 6x^5 * (x - 7y)^2
Step 5: Expand the equation:
x(dy/dx) - 7y(dy/dx) - y + 7y(dy/dx) = 6x^5 * (x - 7y)^2
Step 6: Combine like terms:
x(dy/dx) = 6x^5 * (x - 7y)^2 + y
Step 7: Substitute the coordinates of the given point (1, 6/43) into the equation:
(1) * (dy/dx) = 6(1)^5 * (1 - 7(6/43))^2 + 6/43
Step 8: Simplify and solve for dy/dx, which represents the slope of the tangent line:
dy/dx = (6(1 - 7(6/43))^2 + 6/43) / 1
Step 9: Calculate the value:
dy/dx = 192/43
The slope of the tangent line to the curve at the point (1, 6/43) is 192/43.