Differentiation
12 Toán
June 2024
1 Definition
1.1 Secant line
If (c, f (c)) is the point of tangency and (c + ∆x, f (c + ∆x)) is a second point on the graph of f , the slope
of the secant line through the two points is displayed through the slope formula
f (c + ∆x) − f (c)
msec =
(c + ∆x) − c
f (c + ∆x) − f (c)
= .
∆x
The right-hand side of this equation is a difference quotient. The denominator ∆x is the change in x, and
the numerator ∆y = f (c + ∆x) − f (c) is the change in y.
This expression is called average rate of change.
1.2 Tangent line
The beauty of the given equation is that you can obtain more and more accurate approximations of the slope
of the tangent line by choosing points closer and closer to the point of tangency.
1
,If f is defined on an open interval containing c, and if the limit
∆y f (c + ∆x) − f (c)
lim = lim =m
∆x→0 ∆x ∆x→0 ∆x
exists, then the line passing through (c, f (c)) with slope m is the tangent line to the graph of f at the
point (c, f (c)).
The slope of the tangent line to the graph of f at the point (c, f (c)) is also called the slope of the graph
of f at x = c.
This expression is called instantaneous rate of change.
Example: Find the slopes of the tangent lines to the graph of
f (x) = x2 + 1
at the points (0, 1) and (−1, 2).
Solution:
Let (c, f (c)) represent an arbitrary point on the graph of f . Then the slope of the tangent line at (c, f (c)) is
2
, given by
f (c + ∆x) − f (c) (c + ∆x)2 + 1 − c2 + 1
lim = lim
∆x→0 ∆x ∆x→0 ∆x
c + 2c(∆x) + (∆x)2 + 1 − c2 − 1
2
= lim
∆x→0 ∆x
2c(∆x) + (∆x)2
= lim
∆x→0 ∆x
= lim (2c + ∆x)
∆x→0
= 2c.
So, the slope at any point (c, f (c)) on the graph of f is m = 2c. At the point (0, 1), the slope is m = 2(0) = 0,
and at (−1, 2), the slope is m = 2(−1) = −2.
*Vertical tangent line
If f is continuous at c and
f (c + ∆x) − f (c) f (c + ∆x) − f (c)
lim =∞ or lim = −∞
∆x→0 ∆x ∆x→0 ∆x
the vertical line x = c passing through (c, f (c)) is a vertical tangent line to the graph of f
1.3 Derivative of a function
The derivative of f at x is given by
f (x + ∆x) − f (x)
f ′ (x) = lim
∆x→0 ∆x
provided the limit exists. For all x for which this limit exists, f ′ is a function of x.
This "new" function gives the slope of the tangent line to the graph of f at the point (x, f (x)),
provided that the graph has a tangent line at this point.
*Symmetric numerical differentiation:
Finding values of derivatives using the formula
f (x + ∆x) − f (x − ∆x)
f ′ (x) ≈
2∆x
where ∆x is a small number such as 0.001.
The process of finding the derivative of a function is called differentiation.
A function is differentiable at x if its derivative exists at x and is differentiable on an open interval
(a, b) if it is differentiable at every point in the interval.
In addition to f ′ (x), which is read as " f prime of x," other notations are used to denote the derivative of
y = f (x). The most common are
dy d
f ′ (x), , y′ , [f (x)], Dx [y]
dx dx
3
12 Toán
June 2024
1 Definition
1.1 Secant line
If (c, f (c)) is the point of tangency and (c + ∆x, f (c + ∆x)) is a second point on the graph of f , the slope
of the secant line through the two points is displayed through the slope formula
f (c + ∆x) − f (c)
msec =
(c + ∆x) − c
f (c + ∆x) − f (c)
= .
∆x
The right-hand side of this equation is a difference quotient. The denominator ∆x is the change in x, and
the numerator ∆y = f (c + ∆x) − f (c) is the change in y.
This expression is called average rate of change.
1.2 Tangent line
The beauty of the given equation is that you can obtain more and more accurate approximations of the slope
of the tangent line by choosing points closer and closer to the point of tangency.
1
,If f is defined on an open interval containing c, and if the limit
∆y f (c + ∆x) − f (c)
lim = lim =m
∆x→0 ∆x ∆x→0 ∆x
exists, then the line passing through (c, f (c)) with slope m is the tangent line to the graph of f at the
point (c, f (c)).
The slope of the tangent line to the graph of f at the point (c, f (c)) is also called the slope of the graph
of f at x = c.
This expression is called instantaneous rate of change.
Example: Find the slopes of the tangent lines to the graph of
f (x) = x2 + 1
at the points (0, 1) and (−1, 2).
Solution:
Let (c, f (c)) represent an arbitrary point on the graph of f . Then the slope of the tangent line at (c, f (c)) is
2
, given by
f (c + ∆x) − f (c) (c + ∆x)2 + 1 − c2 + 1
lim = lim
∆x→0 ∆x ∆x→0 ∆x
c + 2c(∆x) + (∆x)2 + 1 − c2 − 1
2
= lim
∆x→0 ∆x
2c(∆x) + (∆x)2
= lim
∆x→0 ∆x
= lim (2c + ∆x)
∆x→0
= 2c.
So, the slope at any point (c, f (c)) on the graph of f is m = 2c. At the point (0, 1), the slope is m = 2(0) = 0,
and at (−1, 2), the slope is m = 2(−1) = −2.
*Vertical tangent line
If f is continuous at c and
f (c + ∆x) − f (c) f (c + ∆x) − f (c)
lim =∞ or lim = −∞
∆x→0 ∆x ∆x→0 ∆x
the vertical line x = c passing through (c, f (c)) is a vertical tangent line to the graph of f
1.3 Derivative of a function
The derivative of f at x is given by
f (x + ∆x) − f (x)
f ′ (x) = lim
∆x→0 ∆x
provided the limit exists. For all x for which this limit exists, f ′ is a function of x.
This "new" function gives the slope of the tangent line to the graph of f at the point (x, f (x)),
provided that the graph has a tangent line at this point.
*Symmetric numerical differentiation:
Finding values of derivatives using the formula
f (x + ∆x) − f (x − ∆x)
f ′ (x) ≈
2∆x
where ∆x is a small number such as 0.001.
The process of finding the derivative of a function is called differentiation.
A function is differentiable at x if its derivative exists at x and is differentiable on an open interval
(a, b) if it is differentiable at every point in the interval.
In addition to f ′ (x), which is read as " f prime of x," other notations are used to denote the derivative of
y = f (x). The most common are
dy d
f ′ (x), , y′ , [f (x)], Dx [y]
dx dx
3
