Find the exact gradient of the curve y=ln(1-cos2x) at the point with x-coordinate π/6

This is a past paper question for an A level OCR MEI paper for Maths.

We need to find the gradient of the curve so we know right away that we need to use differentiation.

The equation y = ln(1-cos2x) is difficult to differentiate by itself so we use the chain rule and a substitution. We say y = ln(u) where u = 1-cos2x.

dy/du = 1/u. This is something you just have to learn, that the differential of y= lnx is 1/x.

If you're stuck with finding du/dx, you can use another substitution if you want where u = 1-cosv and v = 2x, but most people know (after a lot of practice) that the differential of y=coskx is -ksinkx. I believe it's also in most formula books.

So here du/dx = -2sin2x (remember the change of sign when going from cos to sin when differentiating).

So from the chain rule, we know that dy/dx = dy/du x du/dx so dy/dx = 1/u x -2sin2x = -2sin2x/u and u = 1-cos2x so dy/dx = -2sin2x/(1-cos2x)

Now we need to find the value of this gradient when x = π/6

dy/dx = -2sin2x/(1-cos2x) = -2sin(2 x π/6)/(1-cos(2 x π/6)) = -2sin(π/3)/(1-cos(π/3)) 

Remember that we're working in radians here so for this, you need to put your calculator in radians too

We know that cos(π/3) = 1/2 (or 0.5, but 1/2 is easier for the moment) and sin(π/3) = root(3)/2

So dy/dx at x=π/3 gives (-2 x root(3)/2)/(1-1/2) => multiply top and bottom by 2 => -2root(3)/(2-1) = -2root(3)/1 = -2root(3)

Catherine H. A Level Physics tutor, GCSE Chemistry tutor, GCSE Biolog...

12 months ago

Answered by Catherine, an A Level Maths tutor with MyTutor

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


£20 /hr

Jake B.

Degree: Mathematics (Masters) - Durham University

Subjects offered: Maths, Chemistry


“About Me: Hi, my name is Jake, and I am a second year Mathematics Student at Durham University. I have tutored maths for 3 years now, at both GCSE and A level, and I absolutely love it! I My Sessions: During the sessions I will cove...”

£20 /hr

Sarah M.

Degree: Chemistry (Doctorate) - Bristol University

Subjects offered: Maths, Science+ 2 more


“Who am I?I am a French PhD student in chemistry at the university of Bristol. As far as I remember I have always been fascinated by sciences and I ldo ove sharing this passion by helping students.Tutoring is to my eyes extremely re...”

£20 /hr

George B.

Degree: Physics with Industrial experience (Masters) - Bristol University

Subjects offered: Maths, Physics


“Myself: I'm a Physics student at the University of Bristol. My passion is Physics and Maths. I think any theory can be made interesting and exciting. My tutorials will shows my informative and passionate nature.  I have numerous expe...”

About the author

Catherine H.

Currently unavailable: for new students

Degree: Physics with Astrophysics (MSci) (Masters) - Bath University

Subjects offered: Maths, Physics+ 4 more

Further Mathematics

“Hi there! My name's Catherine. I am a first year student at the University of Bath studying Physics and Astrophysics. I spent a year at UPMC, a university in Paris between my A levels and coming to Bath, studying Physics, Chemistry, M...”

You may also like...

Posts by Catherine

Find the exact gradient of the curve y=ln(1-cos2x) at the point with x-coordinate π/6

The speed of water moving through a turbine is 2.5 m/s. Show that the mass of water passing through an area of 500 metres squared in one second is about 1 x 10^6 kg (density of sea water = 1030 kg/m^3)

Other A Level Maths questions

The triangle ABC is such that AC=8cm, CB=12cm, angle ACB=x radians. The area of triangle ABC = 20cm^2. Show that x=0.430 (3sf)

The curve C has equation y = x^3 - 2x^2 - x + 9, x > 0. The point P has coordinates (2, 7). Show that P lies on C.

How do we differentiate y=a^x when 'a' is an non zero real number

Prove that the d(tan(x))/dx is equal to sec^2(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