Find dy/dx by implicit differentiation.
x^3+8x+x^13y-y^6=4
dy/dx = 3x^2+8+13x^12y / 6y^5-x^13
3x^2 + 8 + 13x^12y + x^13y' - 6y^5y' = 0
now just collect terms and solve for y'
To find dy/dx by implicit differentiation, we'll differentiate both sides of the equation with respect to x, treating y as a function of x.
Let's differentiate each term step by step:
1. Differentiate x^3 with respect to x: The derivative of x^n (where n is any constant) with respect to x is nx^(n-1). So, the derivative of x^3 with respect to x is 3x^(3-1) = 3x^2.
2. Differentiate 8x with respect to x: 8x is a simple linear term, so its derivative is just 8.
3. Differentiate x^13y with respect to x: This is a product of x^13 and y. To differentiate a product, we can use the product rule, which states that (uv)' = u'v + uv'.
Using the product rule:
The derivative of x^13 with respect to x is 13x^(13-1) = 13x^12.
The derivative of y with respect to x is dy/dx.
So, the derivative of x^13y with respect to x is 13x^12 * y + x^13 * dy/dx.
4. Differentiate -y^6 with respect to x: This is a power of y, so we'll apply the chain rule. The chain rule states that d(u^n)/dx = n * u^(n-1) * du/dx.
Using the chain rule:
The derivative of -y^6 with respect to x is -6 * y^(6-1) * dy/dx = -6y^5 * dy/dx.
Now, equating the derivatives of each term to zero, we have:
3x^2 + 8 + 13x^12 * y + x^13 * dy/dx - 6y^5 * dy/dx = 0
Next, we'll gather the terms involving dy/dx on one side:
x^13 * dy/dx - 6y^5 * dy/dx = -3x^2 - 8 - 13x^12 * y
Factor out dy/dx:
dy/dx (x^13 - 6y^5) = -3x^2 - 8 - 13x^12 * y
Finally, we can solve for dy/dx by dividing both sides by (x^13 - 6y^5):
dy/dx = (-3x^2 - 8 - 13x^12 * y) / (x^13 - 6y^5)
And there you have it! The derivative dy/dx is given by the equation (-3x^2 - 8 - 13x^12 * y) / (x^13 - 6y^5).