Page 34 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

y y = f(x)
f(x) = x 3
(function of the curve)
t 2
tangent O t 1 x dy = f'(x)
dx
3. The gradient of the tangents at the moment t (first derivative of the function of the curve)
1
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and t is the changes of f(x) as the result of the
2
changes in time which is the speed that is shown 7. The use of the idea of limit in deriving the first
through the motion of the object at the instance derivative of a function f(x) is known as the
t or t . differentiation through first principle.
1 2
4. The gradient of the tangent which happens at the
instance t and t can be determined through the 2
2
1
idea of limit, that is: Determine the first derivative of each of the following
y function f(x) through first principle.
(a) f(x) = 2x (b) f(x) = 5x
2
Q (x y )
chord 2, 2 (c) f(x) = 3x – 8x (b) f(x) = 6
2
x
P tangent Solution
(x y )
1, 1
O x 1 x (a) Let y = f(x), Assuming that x
(t )
1 that is y = 2x experiences a small
Observe that when the point Q is approaching y + dy = 2(x + dx) change (dx) which
point P, the chord PQ is gradually approaching causes a small
and overlapping with the tangent at point P. 2x + dy = 2x + 2dx change (dy) in y.
dy = 2dx
Through the idea of limit, the gradient of tangent dy = 2 Simplify until dy is
dx
of the curve at point P, m = lim y – y 1 . dx obtained by substituting
2
x → x
2 1 x – x 1 lim dy y = 2x.
2
Assuming dx = x – x and dy = y – y , \ dx → 0 dx = 2
1
1
2
2
when x → x , dx → 0. and dy = 2 Use the idea of limit so
that the first derivative of
2
1
y – y
lim
\ m = x → x 1 x – x 1 1 dx function f(x) is obtained.

2
2
2
= lim dy (b) Let y = f(x), 2
that is y = 5x

dx → 0 dx
y + dy = 5(x + dx)
2
SPM Tips 5x + dy = 5[x + 2xdx + (dx) ]
2
2
2
5x + dy = 5x + 10xdx + 5(dx)
2
2
2
dx and dy is read as “delta x” and “delta y”.
dy = 10xdx + 5(dx) Form 5
2
dy = 10x + 5dx
dy dx
5. The symbol is used to represent the gradient
dx lim dy
m which is obtained through the idea of limit \ dx → 0 dx = 10x + 0
and is known as gradient function, that is dy
and = 10x
dx
dy = lim dy
dx dx → 0 dx REMEMBER!
2
6. The gradient of function dy is the first derivative Be careful with the expansion of (x + dx) . Observe that
dx ≠ (dx) .
2
2
dx
of function f(x), that is
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