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
cyan magenta yellow black
(^05255075950525507595)
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(^05255075950525507595)
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\346CamAdd_13.cdr Tuesday, 7 January 2014 2:37:24 PM BRIAN