Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy.
y sec(x) = 9x tan(y)
I will be happy to critique your work on this.
To find dx/dy using implicit differentiation, we need to differentiate both sides of the equation with respect to y, treating x as a function of y.
Let's start by differentiating the left side of the equation y sec(x) = 9x tan(y) with respect to y.
d/dy(y sec(x)) = d/dy(9x tan(y))
To differentiate y sec(x) with respect to y, we apply the chain rule.
The derivative of y sec(x) with respect to y is sec(x) * dy/dy + y * d(sec(x))/dy.
Since dy/dy is just 1, the first term simplifies to sec(x).
Now, let's differentiate 9x tan(y) with respect to y.
The derivative of 9x tan(y) with respect to y is 9x * d(tan(y))/dy + tan(y) * d(9x)/dy.
To differentiate tan(y) with respect to y, we apply the chain rule again.
The derivative of tan(y) with respect to y is sec^2(y) * dy/dy, which simplifies to sec^2(y).
Differentiating 9x with respect to y gives us 0 since x is treated as a constant in this context.
Combining the above results, we have:
sec(x) + y * d(sec(x))/dy = 9x * sec^2(y)
To find dx/dy, solve the above equation for d(sec(x))/dy:
y * d(sec(x))/dy = 9x * sec^2(y) - sec(x)
Next, divide both sides of the equation by y:
d(sec(x))/dy = (9x * sec^2(y) - sec(x)) / y
Now we have an expression for d(sec(x))/dy. However, our goal is to find dx/dy.
To find dx/dy, we need to solve for dx/dy using the above expression and some trigonometric identities.
First, recall that sec(x) is the reciprocal of cos(x):
sec(x) = 1 / cos(x)
Substituting this into the expression for d(sec(x))/dy, we have:
d(sec(x))/dy = (9x * sec^2(y) - 1 / cos(x)) / y
Note that cos(x) can be expressed in terms of sin(x) using the Pythagorean identity:
cos^2(x) + sin^2(x) = 1
Rearranging this equation, we get:
cos^2(x) = 1 - sin^2(x)
Taking the reciprocal of both sides:
1 / cos^2(x) = 1 / (1 - sin^2(x))
Substituting this into the expression for d(sec(x))/dy, we get:
d(sec(x))/dy = (9x * sec^2(y) - 1 / (1 - sin^2(x))) / y
Finally, using the relationship between dx/dy and dy/dx, we can invert this expression to find dx/dy:
dx/dy = 1 / ((9x * sec^2(y) - 1 / (1 - sin^2(x))) / y)