# Given x = 3tan(2y) find dy/dx in terms of x

There are two methods you can use to answer this question. I will briefly outline the second method at the end, as one may seem more intuitive to you than the other.

**Method 1:**

We are asked to find dy/dx, but we have been given a funtion of x in terms of y, instead of a funtion in terms of x, which you will be more used to. (I.e. we have f(y) here instead of f(x)). So we are going to use the fact that **dy/dx = 1 / (dx/dy)**.

We can easily find dx/dy. We differentiate each term of x = 3tan(2y) with respect to y.

The differential of x with respect to y is dx/dy.

The differential of 3tan(2y) with respect to y is 3*2*sec^{2}(2y), using standard results. (We have multiplied by 2 becuase the differential of sec^{2}(f(y)) is f'(y)*sec^{2}(f(y)) - you have to multiply by the differential of the function 'inside' the sec^{2 }due to the chain rule).

This gives us dx/dy = 6sec^{2}(2y). Now the question requires that our answer is in terms of x, not in terms of y, as we have at the moment. So we need to find a **relation** between our answer sec^{2}(2y) and x, where x = 3tan(2y) as in the question. We need to use the identity **tan ^{2}(2y) + 1 = sec^{2}(2y)**. Therefore, if we substitute this in for sec

^{2}(2y), we get

dx/dy = 6*(1 + tan^{2}(2y))

We still need to somehow get this in terms of x. We know x = 3tan(2y). Dividing by 3 means tan(2y) = x/3. Therefore tan^{2}(2y) = x^{2}/9 (just by squaring both sides).

So now we can say that dx/dy = 6*(1 + x^{2}/9), which is now in terms of x!

We need dy/dx, and we have dx/dy. So we take our answer dx/dy and divide 1 by it, as shown by the rule in bold earlier. So dy/dx = 1/(6*(1 + x^{2}/9)).

Taking out a factor of 1/9 in the denominator gives us dy/dx = 1/ ((6/9)*(9 + x^{2})). 1/(6/9) = 9/6 = 3/2.

Therefore our final answer is

dy/dx = 3/ (2*(9 + x^{2}))

**Method 2: Harder if you have difficulties with implicit differentiation**

This method uses implicit differentiation. Differentiating x = 3tan(2y) implicitly gives 1 = 3*2*sec^{2}(2y)*dy/dx. (I have differentiated every term with respect to x, and therefore needed to use the chain rule on the tan(2y). This is where the dy/dx comes from). Rearranging to get dy/dx as the subject gives dy/dx = 1/ 6*(sec^{2}(2y)), and from here you continue in exactly the same way as in method 1.