1132 views

### 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)

2 years ago

Answered by Catherine, an A Level Maths tutor with MyTutor

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

#### 423 SUBJECT SPECIALISTS

£20 /hr

Degree: Physics (Bachelors) - Imperial College London University

Subjects offered:Maths, Physics+ 3 more

Maths
Physics
Further Mathematics
Chemistry
-Personal Statements-

“Undergraduate Physicist with lots of experience in teaching in different fields and age groups, looking to instil a passion for learning into students.”

£24 /hr

Degree: Mathematics (Masters) - Durham University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

£22 /hr

Degree: MMath Pure Mathematics (Masters) - St. Andrews University

Subjects offered:Maths, English Literature

Maths
English Literature

“I am an experienced mathematician with a personal approach to tutoring. I'm here to help you further your mathematical potential.”

£22 /hr

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

Subjects offered:Maths, Physics+ 4 more

Maths
Physics
Further Mathematics
French
Chemistry
Biology

“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

A line L is parallel to y=4x+5 and passes through the point (-1, 6). Find the equation of the line L in the form y=ax+b . Find also the coordinates of its intersections with the axes.

How do you find the minimum of the equation sin^2(x) + 4sin(x)?

Find dy/dx of the equation (x^3)*(y)+7x = y^3 + (2x)^2 +1 at point (1,1)

How would I differentiate cos(2x)/x^1/2

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this.