Cambridge Additional Mathematics

Introduction to differential calculus (Chapter 13) 361

3 Use the laws of logarithms to help differentiate with respect tox:

a y=ln

p

1 ¡ 2 x b y=ln

³ 1

2 x+3

́

c y=ln(ex

p

x) d y=ln(x

p

2 ¡x)

e y=ln

³x+3

x¡ 1

́

f y=ln

μ

x^2

3 ¡x

¶

g f(x) = ln ((3x¡4)^3 ) h f(x) = ln (x(x^2 + 1))

Discovery 9 Derivatives of sinx and cosx

Our aim is to use a computer demonstration to investigate the derivatives ofsinxandcosx.

What to do:

1 Click on the icon to observe the graph of y= sinx. A tangent withx-step of length

1 unit moves across the curve, and itsy-step is translated onto the gradient graph.

Predict the derivative of the function y= sinx.

2 Repeat the process in 1 for the graph of y= cosx. Hence predict the derivative of

the function y= cosx.

From theDiscoveryyou should have deduced that:

Forxin radians: If f(x) = sinx then f^0 (x) = cosx.

If f(x) = cosx then f^0 (x)=¡sinx.

K

DEMO

DERIVATIVES

DEMO

DERIVATIVES OF

TRIGONOMETRIC FUNCTIONS

i f(x)=ln

μ

x^2 +2x

x¡ 5

¶

4 Find the gradient of the tangent to:

a y=xlnx at the point where x=e b y=ln

³x+2

x^2

́

at the point where x=1.

5 Suppose f(x)=aln(2x+b) where f(e)=3and f^0 (e)=

6

e

. Find the constantsaandb.

InChapter 9we saw that sine and cosine curves arise naturally from motion in a circle.

Click on the icon to observe the motion of point P around the unit circle. Observe the graphs

of P’s height relative to thex-axis, and then P’s horizontal displacement from they-axis. The

resulting graphs are those of y= sint and y= cost.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_13\361CamAdd_13.cdr Tuesday, 7 January 2014 11:26:29 AM BRIAN