1. In this chapter, we begin our study of differential calculus.
This is concerned with how one quantity changes in relation to the changes in another quantity.
DERIVATIVE
is a measure of how fast does a function change in response to changes in independent variable;

2. The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways.
One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.
The derivative measures the instantaneous rate of change of the function, as distinct from its average rate of change.

3. Definition of INFINITESIMAL
1: taking on values arbitrarily close to but greater than zero
2: immeasurably or incalculably small
<an infinitesimal difference>
Now let us observe:

4. 𝑥 Gradient/slope of secant and tangent line
A secant is a line that intersect a curve.
A tangent to a curve at a specific point is a straight line that touches the curve at that point.
Slope of the secant line joining 𝑃(𝑎, 𝑓(𝑎)) and 𝑇 𝑎+∆𝑥,𝑓 𝑎+∆𝑥
𝑓 𝑎+∆𝑥 −𝑓(𝑎) ∆𝑥
𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡/𝑠𝑙𝑜𝑝𝑒 =
Let us consider Δ x intervals that are getting smaller and smaller.
As point T approaches point P, secant line becomes tangent line. P T 𝑎
𝑎+∆𝑥
𝑓(𝑥)
f (a+Δx) - f(a) 𝑥

5. 𝐿𝑒𝑡 𝑦= 𝑥 2 −3. 𝐿𝑒𝑡 𝑷 𝑏𝑒 1,−2 .
Gradient of secant line through points 1,−2 and 𝑎+∆𝑥, 𝑓 𝑎+∆𝑥 𝑖𝑠
𝑓 𝑎+∆𝑥 −𝑓(𝑎) ∆𝑥 = 𝑓 1+∆𝑥 −𝑓(1) ∆𝑥
𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
= [(1+∆𝑥) 2 −3]−( 1 2 −3) ∆𝑥 = 2∆𝑥+ (∆𝑥) 2 ∆𝑥
𝑠𝑒𝑐𝑎𝑛𝑡 𝑙𝑖𝑛𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 2∆𝑥+ (∆𝑥) 2 ∆𝑥 𝐟𝐨𝐫 ∀ ∆𝒙
Let Δ𝑥→0, starting with Δ𝑥=1. ∆𝑥
2∆𝑥+ (∆𝑥) 2
2∆𝑥+ (∆𝑥) 2 ∆𝑥 1 3
0.5
1.25
2.5
0.3
0.09
2.3
0.1
0.21
2.1
0.01
0.0201
2.01
0.001
2.001
0.0001
2.0001
As end point (2,1) approaches point (1,-2)
secant line approaches tangent line at (1,-2)
Slopes of the secant lines connecting point (1,-2) and other points on the graph are approaching the slope of the line tangent at (1,-2). In that process both, Δf and Δx are becoming infinitesimally small approaching zero, but their ratio is approaching definite value = slope of tangent line at (1,-2) = 2
𝑙𝑖𝑚 ∆𝑥→0 𝑓 𝑎+∆𝑥 −𝑓(𝑎) ∆𝑥 = 𝑙𝑖𝑚 ∆𝑥→0 [(1+∆𝑥) 2 −3]−( 1 2 −3) ∆𝑥 =2
𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑎𝑡 (1,−2)

6. S Q T R f (a+Δx) - f(a) P f (x) D x 𝑎 x = lim ∆𝑥→0 𝑓 𝑎+∆𝑥 −𝑓(𝑎) ∆𝑥
Df = f (a+Δx) - f(a)
POINTS: 𝑎,𝑓 𝑎 𝑎𝑛𝑑 (𝑎+∆𝑥, 𝑓(𝑎+∆𝑥))
D x x
f (x) S Q T R
f (a+Δx) - f(a) P
D x
Slopes of the secant lines connecting point P
and other points on the graph are approaching
the slope of the line tangent at P as T approaches P.
In that process both, Δ f and Δ x are becoming infinitesimally small approaching zero, but their ratio
is approaching definite value = slope of tangent line at P 𝑎
= lim ∆𝑥→0 𝑓 𝑎+∆𝑥 −𝑓(𝑎) ∆𝑥
𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑎𝑡 𝑃

8. Definition of Derivative
Very often we write
ℎ instead of ∆𝑥
and 𝑓 ′ (a)
Derivative of the function 𝑦=𝑓(𝑥) at a fixed point 𝑥=𝑎:
𝑑𝑓 𝑑𝑥 𝑥=𝑎 = 𝑙𝑖𝑚 ∆𝑥→0 ∆𝑦 ∆𝑥 = 𝑙𝑖𝑚 ∆𝑥→0 𝑓 𝑎+∆𝑥 −𝑓(𝑎) ∆𝑥
𝑓 ′ 𝑎 = 𝑙𝑖𝑚 ℎ→0 𝑓 𝑎+ℎ −𝑓(𝑎) ℎ
Let’s assume that 𝑎 can take any value of 𝑥 on an open interval 𝐼
The derivative of 𝑓 is the function 𝑑𝑓 𝑑𝑥 whose value at any 𝑥∈𝐼 is
𝑑𝑓 𝑑𝑥 = 𝑙𝑖𝑚 ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥
provided that this limit exists.
If this limit exists for each x in an open interval 𝐼, then we say that f is differentiable on 𝐼 (has a derivative everywhere in its domain).

9. Graphycally: If function has derivative at every point in the domain it is differentiable on the domain.

10. 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒
example:
𝑦= 𝑥 2 −3
𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑎𝑡 (𝑥,𝑓 𝑥 )
𝑓′(𝑥)= lim ℎ→0 (𝑥 2 +2𝑥ℎ+ ℎ 2 )− 𝑥 2 ℎ
𝑓 ′ 𝑥 ≡𝑦′= 2𝑥 𝑥
𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒
𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝑥=2𝑥 1 2 -1 -2 4 -4
Equation of the tangent line to a function 𝑦=𝑓(𝑥) at the point 𝑎, 𝑓(𝑎)
𝑦−𝑓 𝑎 = 𝑓 ′ 𝑎 (𝑥−𝑎)

11. Equivalence of Leibnitz’s and Newton’s definition of derivative
Physical: rate of change
Geometrical: Slope of the Tangent to a Curve
The derivative of a function 𝒚=𝒇(𝒙)
with respect to 𝒙 is defined as
The slope of the tangent to a curve 𝒚=𝒇 𝒙 with respect to 𝒙 is defined as
𝑑𝑦 𝑑𝑥 = 𝑙𝑖𝑚 ∆𝑥→0 ∆𝑦 ∆𝑥 = 𝑙𝑖𝑚 ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥
𝑠𝑙𝑜𝑝𝑒= 𝑙𝑖𝑚 ∆𝑥→0 ∆𝑦 ∆𝑥 = 𝑙𝑖𝑚 ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥
The derivative measures the instantaneous rate of change of the function, 𝑑𝑦 𝑑𝑥 , as distinct from its average rate of change, ∆𝑦 ∆𝑥 . P P
f (x)
f (x) x x 𝑥 𝑥
In differentiation Leibniz used the symbols 𝑑𝑥 and 𝑑𝑦 to represent "infinitely small" (or infinitesimal) increments of 𝑥 and 𝑦, just as Δ𝑥 and Δ𝑦 represent finite increments of 𝑥 and 𝑦.[
The ratio of two finite increments becomes the ratio of two infinitesimal increments in the process of finding the limit, yet still that ratio is a finite value. And it is equal to the slope of tangent line.

12. 𝑦 ′ 𝑥 𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒 𝑦 = 𝑑𝑦 𝑑𝑡 𝑁𝑒𝑤𝑡𝑜𝑛
Notation:
𝑑𝑦 𝑑𝑥 𝐿𝑒𝑖𝑏𝑛𝑖𝑡𝑧
𝑦 ′ 𝑥 𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒 𝑦 = 𝑑𝑦 𝑑𝑡 𝑁𝑒𝑤𝑡𝑜𝑛
When derivatives are taken with respect to time, they are often denoted using Newton's dot notation
There are many ways to write and read the derivative of
𝑓 ′ 𝑥 "𝑓 𝑝𝑟𝑖𝑚𝑒 𝑥" 𝑜𝑟 "𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑥"
𝑦 ′ "𝑦 𝑝𝑟𝑖𝑚𝑒 𝑥" 𝑜𝑟 "𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑦 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑥"
𝑑𝑦 𝑑𝑥 "𝑑𝑦 𝑑𝑥" 𝑜𝑟 "𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑦 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑥"
𝑑𝑓 𝑑𝑥 "𝑑𝑓 𝑑𝑥" 𝑜𝑟 "𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑥"
𝑑 𝑑𝑥 𝑓(𝑥) "𝑑 𝑑𝑥 𝑜𝑓 𝑓 𝑜𝑓 𝑥" 𝑜𝑟 "𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑥"

13. Alternate limit form of derivative
𝑓 ′ 𝑎 = 𝑙𝑖𝑚 ℎ→0 𝑓 𝑎+ℎ −𝑓(𝑎) 𝑎+ℎ −𝑎
𝑓 ′ 𝑎 = 𝑙𝑖𝑚 𝑥→𝑎 𝑓 𝑥 −𝑓(𝑎) 𝑥−𝑎

14. Differentiability
A function 𝑓(𝑥) is differentiable on an open interval 𝐼 (has a derivative) if the limit
𝑑𝑓 𝑑𝑥 = 𝑙𝑖𝑚 ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥
exists for ∀ 𝑥∈𝐼
Functions on closed intervals must have one-sided derivatives defined at the end points.
To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner
A function has derivative at a point, if the left derivative is equal to the right derivative at that point.
The left slope must be equal to the right slope.
cusp
vertical tangent
discontinuity

15. Differentiability implies continuity,
If f is differentiable at x = c, then f is continuous at x = c.
Since a function must be continuous to have a derivative,
if it has a derivative then it is continuous. ≡
The converse:
"If a function is continuous at c, then it is differentiable at c," - is not true.
This happens in cases where the function "curves sharply."
𝑣𝑖𝑠𝑢𝑎𝑙𝑙𝑦: 𝑙𝑒𝑓𝑡 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒≠𝑟𝑖𝑔ℎ𝑡 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒
Differentiability implies continuity,
continuity doesn’t imply differentiability.

16. example: 𝑥
The derivative is defined at the end points of a function on a closed interval.
The derivative is the slope of the original function. 𝑥

17. Rate of change
The derivative is the instantaneous rate of change of a function with respect its variable. This is equivalent to finding the slope of the tangent line to the function at a point.
This is where Newton’s and Leibnitz approach meet.
Rate of change is large when the derivative is large (and therefore the curve is steep, as at the point P in the figure), the y-values change rapidly.
Rate of change is small when derivative is small, the curve is relatively flat and the y-values change slowly.

18. Although you are going to use certain simple rules to compute derivatives that will greatly simplify the task of differentiation, sometimes there will be a question to find derivative directly from the definition (very good to remind you of the true meaning of derivative).
Before we introduce these simple rule without which calculus BC would be even worse nightmare, there would be no Ipad and the life as we know it ……. let us use definition of derivative, and find some simple rules.

19. 𝑦−𝑓 𝑎 = 𝑓 ′ 𝑎 (𝑥−𝑎) 𝑦−𝑓 𝑎 =− 1 𝑓 ′ 𝑎 (𝑥−𝑎)
Equation of the tangent line to a function 𝑦=𝑓(𝑥) at the point 𝑎, 𝑓(𝑎)
𝑦−𝑓 𝑎 = 𝑓 ′ 𝑎 (𝑥−𝑎)
Equation of the normal line to a function 𝑦=𝑓(𝑥) at the point 𝑎, 𝑓(𝑎)
𝑦−𝑓 𝑎 =− 1 𝑓 ′ 𝑎 (𝑥−𝑎)

20. The hyperbola and its tangent
Find an equation of the tangent line to the hyperbola 𝑦= 3 𝑥 at the point (3, 1).
The slope of the tangent at (3, 1) is:
3 3+ℎ − 1 ℎ
− ℎ ℎ(3+ℎ)
− 1 3+ℎ
1 3
𝑓 3+ℎ −𝑓(3) ℎ
𝑚= lim ℎ→ = lim ℎ→ = lim ℎ→ = lim ℎ→ =−
Eq. of the tangent at the point (3, 1) is 𝑦−1=− 1 3 (𝑥−3)
𝑥 + 3𝑦 – 6 = 0
The hyperbola and its tangent
are shown in the figure
How do you find normal at (3,1) ?

21. 𝐻𝑜𝑤 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑔𝑟𝑎𝑝ℎ 𝑓 ′ 𝑥 𝑘𝑛𝑜𝑤𝑖𝑛𝑔 𝑔𝑟𝑎𝑝ℎ 𝑓(𝑥)
𝐻𝑜𝑤 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑔𝑟𝑎𝑝ℎ 𝑓 ′ 𝑥 𝑘𝑛𝑜𝑤𝑖𝑛𝑔 𝑔𝑟𝑎𝑝ℎ 𝑓(𝑥)
We can estimate the value of the derivative at any value of x by drawing the tangent at the point (𝑥,𝑓(𝑥)) and estimating its slope.
For instance, for x = 5, we draw the tangent at P in the figure
and estimate its slope to be about 3/2, so 𝑓′(5)≈1.5.
This allows us to plot the point 𝑃’(5, 1.5) on the graph of f’ directly beneath P.
Repeating this procedure at several points,
we get the graph shown in this figure.
Tangents at x = A, B, and C are horizontal.
So, the derivative is 0 there and the graph
of f’ crosses the x-axis at those points.
Between A and B, the tangents have positive
slope. So, f’(x) is positive there.
Between B and C, and the tangents have
negative slope. So, f’(x) is negative there.

22. Derivative of a function at a point gives
• The slope of the tangent line at that point
• The instantaneous rate of change at that point
Application of the Derivative to Motion
Let 𝑠 𝑡 be a position function as a function of time 𝑡 and 𝑣(𝑡) be a velocity function as a function of time.
Then:
The average rate of change of position on a time interval from 𝑡=𝑎 to 𝑡=𝑏 is called average velocity.
•𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = ∆𝑠 ∆𝑡 = 𝑠 𝑏 −𝑠 𝑎 𝑏−𝑎 =𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑠𝑒𝑐𝑎𝑛𝑡 𝑙𝑖𝑛𝑒
The instantaneous rate of change of position at time 𝑡 is called instantaneous velocity.
•𝐼𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣 𝑡 = 𝑠 ′ 𝑡 = lim ∆𝑡→0 ∆𝑠 ∆𝑡 =𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒