Use implicit differentiation to find the slope of the tangent line to the curve y/( x + 4 y) = x^4 – 4
at the point (1,-3/13)
To find the slope of the tangent line to the given curve at the point (1, -3/13), we need to use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
Let's differentiate the equation y / (x + 4y) = x^4 - 4 term by term.
First, let's differentiate the left side. We will use the quotient rule:
(d/dx)(y / (x + 4y)) = [(d/dx)(y)(x + 4y) - y (d/dx)(x + 4y)] / (x + 4y)^2
Now, let's differentiate the right side.
(d/dx)(x^4 - 4) = 4x^3
Step 2: Simplify the equation and solve for dy/dx.
Substituting the derivatives into our equation, we have:
[(d/dx)(y)(x + 4y) - y (d/dx)(x + 4y)] / (x + 4y)^2 = 4x^3
Now, let's solve for dy/dx, which represents the slope of the tangent line.
[(dy/dx)(x + 4y) + y (d/dx)(x + 4y)] / (x + 4y)^2 = 4x^3
We can simplify further by expanding and rearranging the terms:
[(x + 4y)(dy/dx) + y (1 + 4(dy/dx))] / (x + 4y)^2 = 4x^3
Step 3: Substitute the values of x and y from the given point.
Given point: (1, -3/13)
Substituting the values x=1 and y=-3/13 into our equation, we have:
[(1 + 4(-3/13))(dy/dx) + (-3/13)(1 + 4(dy/dx)))] / (1 + 4(-3/13))^2 = 4(1)^3
Simplifying the equation further, we get:
[(1 - 12/13)(dy/dx) + (-3/13)(1 + 4(dy/dx)))] / (1 - 12/13)^2 = 4
Simplifying the denominators further, we have:
[(1/13)(dy/dx) + (-3/13)(1 + 4(dy/dx)))] / (1/13)^2 = 4
Multiplying through by (1/13)^2 to eliminate the denominators, we get:
(dy/dx) - 3(1 + 4(dy/dx))) = 52
Simplifying, we have:
dy/dx - 3 - 12(dy/dx) = 52
Combining like terms, we get:
-11(dy/dx) - 3 = 52
Solving for dy/dx, we have:
-11(dy/dx) = 52 + 3
-11(dy/dx) = 55
dy/dx = -55/11
dy/dx = -5
So, the slope of the tangent line to the curve at the point (1, -3/13) is -5.