Find dy/dx in terms of x and y if xlny+y^6= (9)lnx
To find dy/dx in terms of x and y, we can use implicit differentiation. Implicit differentiation allows us to differentiate both sides of the equation with respect to x, assuming that y is a function of x.
Let's differentiate both sides of the equation:
For the left side, we use the product rule. Let u = xlny and v = y^6. Applying the product rule, we have:
d(uv)/dx = u*(dv/dx) + v*(du/dx)
d(uv)/dx = d(xlny*y^6)/dx = x* d(lny*y^6)/dx + (lny*y^6) * d(x)/dx
For the right side, we use the chain rule. Let's assume that f(x) = ln(x). Using the chain rule, we have:
d(f(x))/dx = d(ln(x))/dx = 1/x
So, applying the chain rule to (9)ln(x), we get:
d((9)lnx)/dx = (9)*(d(lnx)/dx) = (9)*(1/x)
Now, let's substitute these results back into the equation:
x * d(lny*y^6)/dx + (lny*y^6) * 1 = (9) * (1/x)
x * d(lny)/dx * y^6 + x * 6y^5 * dy/dx + lny * y^6 = (9) / x
Now, let's isolate dy/dx:
x * 6y^5 * dy/dx = -(x * d(lny)/dx * y^6) + (9) / x - lny * y^6
dy/dx = [-(x * d(lny)/dx * y^6) + (9) / x - lny * y^6] / [x * 6y^5]
Now, we can simplify this expression, which is the derivative dy/dx in terms of x and y.