This document provides information about derivatives and their applications: 1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions. 2. Higher order derivatives are introduced, with examples of how to take second and third derivatives. 3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.

1. DERIVATIVES
BRIDGE COURSE
F.Y Electrical-1
By:-
Nisarg Amin 14BEEEG012
Deep Pandya 14BEEEF011
Urjit Trivedi 14BEEEG008
Pratik Sharma 14BEEE007
Kunj Shah 14BEEEG010

2. Derivatives
The derivative, or derived function of f(x) denoted by f `(x) is defined
as
f x f x h f x
= æ + - ö çè ø¸
`( ) lim ( ) ( )
0
h
® h
f (x + h) - f (x)
h
x x + h
P
Q
x
y
f x y dy
= æ ö = çè ø¸
Leibniz Notation: `( ) lim
h o
d
x dx
® d

3. Differentiation from first principles
Given that f(x) is differentiable, we can use the definition to prove
that if f (x) = x2 then f `(5) =10
f x = f x + h -
f x
`( ) lim ( ) ( )
0
h
® h
2 2
= + -
lim ( )
h
0
x h x
® h
2 2 2
= + + -
lim 2
h
0
x xh h x
® h
2
= +
lim 2
h
0
xh h
® h
= +
lim (2 )
h
0
h x h
® h
= lim(2 +
)
h
0
As h®0, f `(x)®2x
f `(5) =10
x h
®

4. NOTE Not all functions are differentiable.
y
y = tan(x).
x
Here, tan(x) has ‘breaks’ in the graph where
the gradient is undefined
y y = x
Although the graph is continuous, the
derivative at zero is undefined as the left
derivative is negative and the right
derivative is positive.
x
For a function to be differentiable, it must be continuous.

5. The Product Rule
If k(x) = f (x).g(x) , then:
k`(x) = f `(x)g(x) + g`(x) f (x)
Using Leibniz notation,
If y = f (x).g(x), then:
dy df .g(x) dg . f (x)
dx dx dx
= + OR dy f `g g`f
dx
= +

6. 1. Differentiate y = x2 sin x
f (x) = x2 g(x) = sin x
f `(x) = 2x g`(x) = cos x
dy f `g g`f
dx
= +
= 2xsin x + x2 cos x

7. The Quotient Rule
k x f x
If ( ) ( ) , then:
g x
( )
=
k x = f x g x -
g x f x
`( ) `( ) ( ) `( ) ( )
2
g x
( ( ))
Using Leibniz notation,
If y f ( x
) , then:
g x
( )
=
df g x dg f x dy dx dx
dx g x
. ( ) . ( )
( ( ))
2
-
dy = f `g -
g`f
dx g
= OR 2

8. æ 3
ö
ç ¸
è ø
1. Find
d x
dx sin
x
f (x) = x3 g(x) = sin x
f `(x) = 3x2 g`(x) = cos x
dy = f `g -
g`f
dx g
2
2 3
x x x x
3 sin cos
2
sin
x
= -

9. Sec, Cosec, cot and tan
sec 1 the secant of
x x
cos
x
=
cos 1 the cosecant of
ec x x
sin
x
=
cot 1 the cotangent of
x x
tan
x
=
Unlike the sine and cosine functions, the graphs of sec and cosec
functions have ‘breaks’ in them.
The functions are otherwise continuous but for certain values of x,
are undefined.

10. y
x
1
2
3
4
5
6
6
5
4
3
2
1
– 1
– 2
– 3
– 4
– 5
– 6
y = sec x
123456–––––– 123456
y
x
1
1
2
2
3
3
4
4
5
5
6
6
1
1
2
2
3
3
4
4
5
5
6
6
– 1
– 1
2
– 2
3
– 3
4
– 4
5
– 5
6
– 6
y = cosec x
123456–––––– 123456
y
x
1
1
2
2
3
3
4
4
5
5
6
6
1
1
2
2
3
3
4
4
5
5
6
6
– 1
– 1
2
– 2
3
– 3
4
– 4
5
– 5
6
– 6
y = cot x
x x = p + np
In general, sec is undefined for
2
also cosec x is undefined for x = np

11. x x = p + np
In general, sec is undefined for
2
sec 1
æ ö = çè ø¸ æ ö
2 cos
2
p
p
çè ø¸
1
0
= undefined
also cosec x is undefined for x = np

12. 1. Find the derivative of tan x
d ( tan x ) d sin
x
dx dx x
= æ ö çè cos
ø¸
(quotient rule)
x x x x
= - -
cos .cos ( sin ).sin
2
cos
x
1
cos 2
x
=
= sec2 x

13. Exponential and Logarithmic functions
If y = ex then dy =
ex
dx
y x dy x
If = log then = 1 ¹
0 e
dx x

14. 1. Find d e3x
dx
d e3x =
3e3x
dx
2. Find d
dx
xex
d xex ex xex
dx
= + = ex (1+ x)
3. Find d ln3x
dx
ln3 1 .3
d x
dx x
= 1
3
x
=

15. Higher derivatives
Function 1st derivative 2nd Derivative………..nth Derivative
f (x) f '(x) f ''(x) f n (x)
dy
dx
2
2
d y
dx
n
n
d y
dx
y
1. y = 3x4
dy =
12x3
dx
2
2
d y =
36x
dx
2 3
3 d y 72x
dx
= etc. etc. etc.

16. The second derivative, or second order derivative, is the
derivative of the derivative of a function. The derivative
of the function f(x) may be denoted by f’(x) , and its
double (or "second") derivative is denoted by f’’(x) .
This is read as "f double prime of x," or "The second
derivative of f(x)." Because the derivative of function is
defined as a function representing the slope of function ,
the double derivative is the function representing the
slope of the first derivative function.

17. Furthermore, the third derivative is the
derivative of the derivative of the derivative of
a function, which can be represented by
f’’’(x) . This is read as "f triple prime of x", or
"The third derivative of f(x)". This can
continue as long as the resulting derivative is
itself differentiable, with the fourth derivative,
the fifth derivative, and so on. Any derivative
beyond the first derivative can be referred to as
a higher order derivative.

18. Applications of Derivatives
Let displacement from an origin be a function of time.
x = f (t)
Velocity is a rate of change of displacement.
v dx
=
dt
Acceleration is a rate of change of velocity.
a dv
dt
=
2
2
d x
dt
=

19. A particle travels along the x axis such that
x(t) = 4t3 – 2t + 5,
where x represents its displacement in metres from the origin ‘t’
seconds after observation began.
(a) How far from the origin is the particle at the start of
observation?
(b) Calculate the velocity and acceleration of the particle after 3
seconds.
(a) When t = 0, x(0) = 5
Hence the particle is 5m from the origin at the start of the observation.
(b) v = dx = 12t2 - 2
v(3) =12´9 - 2 =106ms-1
dt
2
2 a d x 24t
= = a(3) = 24´3 = 72ms-2
dt

20. Maximum And Minimum
Two types of maxima and minima:
1) Local maximum and minimum
2) Absolute maximum and minimum
Maximum and Minimum are collectively called
EXTREMA.

21. Local max
Local max
Local min
Local &
Absolute
Max
Local &
Absolute

22. Local Maximum and Minimum
Local extrema are the extrema which occur in the
neighborhood of the function.
• How to find local maximum and minimum?
Let f(x) be any function.
1. Assume f(x) = 0
2. Find critical points
3. If f”(x) > 0 then function has local minimum
value at ‘x’.
If f”(x) < 0 then function has local maximum
value at ‘x’.

23. Absolute Maximum and Minimum
Absolute extrema are the largest and smallest values
that a function takes on over its entire domain.
• How to find absolute maximum and minimum?
Let f(x) be any function.
1. Assume f(x) = 0
2. Find critical points
3. Put critical points in f(x).
The value of ‘x’ for which maximum value of f(x) is
obtained is Absolute Maximum.
The value of ‘x’ for which minimum value of f(x) is
obtained is Absolute Minimum.

24. Illustration :-
• Find local and absolute maximum and minimum
3 2
3 2
of x - 6x – 36x + 2.
f(x) = x - 6x – 36x + 2
f’(x) = 3x - 2
12x – 36
f’(x) = 3(x - 2
4x–12)
f’(x) = 3(x-6) (x+2)
f’(x) = 0
3(x-6) (x+2) = 0
x = 6 or x = -2
Critical points = 6,-2

25. Now,
f”(x) = 6x – 12
f”(6) = 36 – 12
= 24 > 0
f”(-2) = -12 – 12
= - 24 < 0
Function has local maximum value at x = -2
Function has local minimum value at x = 6
Also,
f(6) = 216 – 216 - 216 + 2
= -214
f(-2) = -8 – 24 + 72 + 2
= 42
Function has absolute maximum value at x = -2
Function has absolute minimum value at x = 6