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)
I would cross-multiply, then simplify to avoid using the quotient rule to get
36y = x^7 + 5x - 7x^6y
36dy/dx = 7x^6 + 5 -7x^6dy/dx - y(42x^5)
dy/dx(36 + 7x^6) = 7x^6 + 5x - 42yx^5
dy/dx = (7x^6 + 5x - 42yx^5)/(36+7x^6)
sub in x=1 and y = 6/43)
I will leave the arithmetic for you
THANK YOU !
when you cross multiply how do you get 36y?
To find the slope of the tangent line using implicit differentiation, you'll need to follow these steps:
Step 1: Start by differentiating both sides of the equation with respect to 'x'. Treat y as a function of x and apply the chain rule when differentiating terms with y.
The given equation is:
(y) / (x - 7y) = x^6 + 5
Differentiating both sides of the equation with respect to 'x', we get:
d/dx [(y) / (x - 7y)] = d/dx [x^6 + 5]
Step 2: Apply the quotient rule to differentiate the left-hand side of the equation.
The quotient rule states: d(u/v)/dx = (v * du/dx - u * dv/dx) / v^2.
Using the quotient rule, the left-hand side can be differentiated as follows:
[(x - 7y) * dy/dx - y * d(x - 7y)/dx] / (x - 7y)^2 = 6x^5
Simplifying this expression, we have:
[(x - 7y) * dy/dx - y * (1 - 7(dy/dx))] / (x - 7y)^2 = 6x^5
Step 3: Rearrange the equation to solve for dy/dx, which represents the derivative of y with respect to x.
Expanding the numerator, the equation becomes:
[x * dy/dx - 7y * dy/dx - y + 7y * dy/dx] / (x - 7y)^2 = 6x^5
Combining like terms, the equation simplifies to:
x * dy/dx - y / (x - 7y) = 6x^5
Step 4: Substitute the given point (1, 6/43) into the equation.
Substituting x = 1 and y = 6/43 into the equation, we get:
1 * dy/dx - (6/43) / (1 - 7(6/43)) = 6(1^5)
dy/dx - (6/43) / (1 - (42/43)) = 6
Simplifying, we have:
dy/dx - (6/43) / (1 - 42/43) = 6
dy/dx - (6/43) / (1/43) = 6
dy/dx - 6 = 6
Step 5: Solve for dy/dx, which represents the slope of the tangent line at the given point.
Adding 6 to both sides of the equation, we get:
dy/dx = 12
Therefore, the slope of the tangent line to the curve (y) / (x - 7y) = x^6 + 5 at the point (1, 6/43) is 12.