Cambridge Additional Mathematics

346 Introduction to differential calculus (Chapter 13)

Example 5 Self Tutor

If y=3x^2 ¡ 4 x, find

dy

dx

and interpret its meaning.

As y=3x^2 ¡ 4 x,

dy

dx

=6x¡ 4.

dy

dx

is: ² the gradient function or derivative of y=3x^2 ¡ 4 x from which the gradient of

the tangent at any point on the curve can be found

² the instantaneous rate of change ofywith respect tox.

Example 6 Self Tutor

Find f^0 (x) for f(x) equal to:

a 5 x^3 +6x^2 ¡ 3 x+2 b 7 x¡^4

x

+^3

x^3

a f(x)=5x^3 +6x^2 ¡ 3 x+2

)f^0 (x) = 5(3x^2 ) + 6(2x)¡3(1)

=15x^2 +12x¡ 3

b f(x)=7x¡

4

x

+

3

x^3

=7x¡ 4 x¡^1 +3x¡^3

)f^0 (x) = 7(1)¡4(¡ 1 x¡^2 )+3(¡ 3 x¡^4 )

=7+4x¡^2 ¡ 9 x¡^4

=7+

4

x^2

¡

9

x^4

Example 7 Self Tutor

Find the gradient function of y=x^2 ¡

4

x

and hence find the gradient of the tangent to the function

at the point where x=2.

y=x^2 ¡

4

x

=x^2 ¡ 4 x¡^1

)

dy

dx

=2x¡4(¡ 1 x¡^2 )

=2x+4x¡^2

=2x+

4

x^2

When x=2,

dy

dx

=4+1=5.

So, the tangent has gradient 5.

Remember that

.

_______________

_______________________

1

^^xn=x-n

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\346CamAdd_13.cdr Tuesday, 7 January 2014 2:37:24 PM BRIAN