Question

Use implicit differentiation to find the slope of the tangent line to the curve (y)/(x - 3y) = x^6 - 7 at the point (1, 6/17).

Answer 1

To find the slope of the tangent line to the curve, we can use implicit differentiation. Differentiate both sides of the equation with respect to x. Substitute the given point into the equation to find the value of dy/dx at that point.

Step-by-step explanation:

To find the slope of the tangent line to the curve, we can use implicit differentiation. Let's differentiate both sides of the equation with respect to x.

First, we differentiate the left side using the quotient rule:

d/dx(y/(x - 3y)) = d/dx(x^6 - 7)

Using the quotient rule, the derivative of the left side is:

(x - 3y)*(dy/dx) - y*(1 - 3(dy/dx)) / (x - 3y)^2 = 6x^5

Simplifying this equation, we get:

x*(dy/dx) - 3y*(dy/dx) + y = 6x^5*(x - 3y)^2

Now, we substitute the given point (1, 6/17) into the equation to find the value of dy/dx at that point:

1*(dy/dx) - 3*(6/17)*(dy/dx) + 6/17 = 6*(1^5)*(1 - 3*(6/17))^2

Solving for dy/dx, we find that the slope of the tangent line at the point (1, 6/17) is:

dy/dx = 1/3