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*sec2(2y), using standard results. (We have multiplied by 2 becuase the differential of sec2(f(y)) is f'(y)*sec2(f(y)) - you have to multiply by the differential of the function 'inside' the sec2 due to the chain rule). 

This gives us dx/dy =  6sec2(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 sec2(2y) and x, where x = 3tan(2y) as in the question. We need to use the identity tan2(2y) + 1 = sec2(2y). Therefore, if we substitute this in for sec2(2y), we get

dx/dy = 6*(1 + tan2(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 tan2(2y) = x2/9 (just by squaring both sides). 

So now we can say that dx/dy = 6*(1 + x2/9), 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 + x2/9)).

Taking out a factor of 1/9 in the denominator gives us dy/dx = 1/ ((6/9)*(9 + x2)). 1/(6/9) = 9/6 = 3/2. 

Therefore our final answer is

dy/dx = 3/ (2*(9 + x2))

Method 2: Harder if you have difficulties with implicit differentiation

This method uses implicit differentiation. Differentiating x = 3tan(2y) implicitly gives 1 = 3*2*sec2(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*(sec2(2y)), and from here you continue in exactly the same way as in method 1. 

Annabel W. GCSE Maths tutor, A Level Maths tutor, GCSE Chemistry tutor

8 months ago

Answered by Annabel, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

James B.

Degree: Mathematics (Masters) - Exeter University

Subjects offered: Maths


“About MeI'm a second year student at Exeter University. I've been passionate about maths from a young age and have carried that love through my school and uni career. I hope that I can generate the same passion for my tutees as well....”

£20 /hr

Caroline H.

Degree: Chemistry (Masters) - Edinburgh University

Subjects offered: Maths, Chemistry+ 1 more


“I am a chemistry student at Edinburgh University. Although I have achieved highly I have always had to put in a little more effort than most going through material just that extra time. I am keen to be able to offer others the support ...”

£36 /hr

Jacan C.

Degree: Theoretical Physics (Masters) - York University

Subjects offered: Maths, Science+ 3 more

Further Mathematics

“Helping the striving and the struggling with caring, friendly and structured tuition. With 200+ hours of experience, offering Physics, Maths and Chemistry.”

About the author

Annabel W.

Currently unavailable: for regular students

Degree: Mathematics (Masters) - Durham University

Subjects offered: Maths, Chemistry


“Hi! I'm Annie, a first year Mathematics student studying a Durham University. If you're looking for support in your studies in Maths or Chemistry, I'm here to help! I'm very enthusiastic about helping others to learn and progress, and ...”

You may also like...

Posts by Annabel

Explain what is meant by the term isotope

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

Solve 3x^2 = 8x - 2 giving your answers to 2 d.p.

Other A Level Maths questions

The equation of a curve is x(y^2)=x^2 +1 . Using the differential, find the coordinates of the stationary point of the curve.

If n is an integer prove (n+3)^(2)-n^(2) is never even.

Solve the inequality x < 4 - |2x + 1|.

Find the integral of 3x^2 + 4x + 9 with respect to x.

View A Level Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss