To evaluate higher order derivative of trigonometric functions like the tangent function, we must become familiar with all of the other derivation rules; in this case we will use the Chain Rule and the Product Rule. 

We begin with the identity that 

Since the first derivative has now turned into a composite function in the form

 where    and   

and      and  

we have that the second derivative, utilizing the Chain Rule, is

To obtain the third derivative, we must derive the second derivative via the Product Rule, since there is the multiplication of u=2[sec(x)]^2 and v=tan(x).

Recall that the Product Rule states that for a function, f(x)=uv:

As a result we have that

In order to solve for the derivative of an equation that is not written as a function of a single variable (x), but rather two variables (x and y), we must implement Implicit Differentiation. 

Implicit Differentiation states that when the derivative of a function in terms of one variable (in this case, y) is taken with respect to the other variable present (in this case,x), one must multiply that derived term, f'(y) by the by the derivative (y' or dy/dx). 

In other words, if we have a function in terms of y, we have that it's derivative with respect to x is 

In this case, we must derive both sides of the equation, so we begin with the left side f(x,y)=6xy. Since two terms are being multiplied, we must implement the product rule. We then have that

Then we look at the right side of the equation f(x,y)=x^2+y^2 and we have that

Putting together the left and the right side of the equations, we have that

Algebraically, we simplify this equation to

To isolate our derivative term (dy/dx), we divide the left term by (2y-6x) to get:

To find the derivative of a function in this form, we must make use of the chain rule, which states that for a give function: 

Its derivative is defined as:

In this case we have the function:

which is in the form of the composite function

in which

 ,     ,     

By similar logic, the derivative of the composite function is 

in which

and  

As a result, we have that 

The Product Rule of derivatives states that for a given function:

The derivative is defined as 

In this case, for the given function

 , 

and the respective derivatives are:

 , 

Applying the product rule we get that

To find the first derivative of this function, we must make use of the Quotient Rule of derivatives. That is, for a function

The derivative is defined as 

In this case 

 , 

 , 

As a result, 

First we note that 

.  

Now we take the derivative  of the function to find the slope of the tangent line.

Using the chain rule we get 

.  

Now we want to find the specific slope of the tangent line containing the .  

We then find that the slope is 

.  

We now plug in these values to slope-intercept form to find the y-intercept of the tangent line. 

which gives us 

.  

Thus the equation is simply 

.