Ilm Gah Website is under maintenance! If you found any error please let us know. Contact Us ×

Derivative Rules


There are rules we can follow to find many derivatives.

The Derivative tells us the slope of a function at any point.
slope examples y=3, slope=0; y=2x, slope=2
There are rules we can follow to find many derivatives. For example:
  • The slope of a constant value (like 3) is always 0
  • The slope of a line like 2x is 2, or 3x is 3 etc
  • and so on.
Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark  means "Derivative of".
Common FunctionsFunctionDerivative
Square Root√x(½)x
axln(a) ax
loga(x)1 / (x ln(a))
Trigonometry (x is in radians)sin(x)cos(x)
Inverse Trigonometrysin-1(x)1/√(1−x2)
Multiplication by constantcfcf’
Power Rulexnnxn−1
Sum Rulef + gf’ + g’
Difference Rulef - gf’ − g’
Product Rulefgf g’ + f’ g
Quotient Rulef/g(f’ g − g’ f )/g2
Reciprocal Rule1/f−f’/f2
Chain Rule
(as "Composition of Functions")
f º g(f’ º g) × g’
Chain Rule (using ’ )f(g(x))f’(g(x))g’(x)
Chain Rule (using ddx )dydx = dydududx
"The derivative of" is also written ddx
So ddxsin(x) and sin(x)’ are the same thing, just written differently


Example: what is the derivative of sin(x) ?

From the table above it is listed as being cos(x)
It can be written as:
d/dxsin(x) = cos(x)
sin(x)’ = cos(x)

Power Rule

Example: What is d/dxx3 ?

The question is asking "what is the derivative of x3?"
We can use the Power Rule, where n=3:
d/dxxn = nxn−1
d/dxx3 = 3x3−1 = 3x2

Example: What is d/dx(1/x) ?

1/x is also x-1
We can use the Power Rule, where n = −1:
d/dxxn = nxn−1
d/dxx−1 = −1x−1−1 = −x−2

Multiplication by constant

Example: What is d/dx5x?

the derivative of cf = cf’
the derivative of 5f = 5f’
We know (from the Power Rule):
d/dxx3 = 3x3−1 = 3x2
d/dx5x3 = 5d/dxx3 = 5 × 3x2 = 15x2

Sum Rule

Example: What is the derivative of x2+x?

The Sum Rule says:
the derivative of f + g = f’ + g’
So we can work out each derivative separately and then add them.
Using the Power Rule:
  • d/dxx2 = 2x
  • d/dxx3 = 3x2
And so:
the derivative of x2 + x3 = 2x + 3x2

Difference Rule

It doesn't have to be x, we can differentiate with respect to, for example, v:

Example: What is d/dv(v3−v4) ?

The Difference Rule says
the derivative of f − g = f’ − g’
So we can work out each derivative separately and then subtract them.
Using the Power Rule:
  • d/dvv3 = 3v2
  • d/dvv4 = 4v3
And so:
the derivative of v3 − v4 = 3v2 − 4v3

Sum, Difference, Constant Multiplication And Power Rules

Example: What is d/dz(5z2 + z3 − 7z4) ?

Using the Power Rule:
  • d/dzz2 = 2z
  • d/dzz3 = 3z2
  • d/dzz4 = 4z3
And so:
d/dz(5z2 + z3 − 7z4) = 5 × 2z + 3z2 − 7 × 4z3 = 10z + 3z2 − 28z3

Product Rule

Example: What is the derivative of cos(x)sin(x) ?

The Product Rule says:
the derivative of fg = f g’ + f’ g
In our case:
  • f = cos
  • g = sin
We know (from the table above):
  • d/dxcos(x) = −sin(x)
  • d/dxsin(x) = cos(x)
the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)

cos2(x) − sin2(x)


Reciprocal Rule

Example: What is d/dx(1/x) ?

The Reciprocal Rule says:
the derivative of 1/f = −f’/f2
With f(x)= x, we know that f’(x) = 1
the derivative of 1/x = −1/x2
Which is the same result we got above using the Power Rule.

Chain Rule

Example: What is ddxsin(x2) ?

sin(x2) is made up of sin() and x2:
  • f(g) = sin(g)
  • g(x) = x2
The Chain Rule says:
the derivative of f(g(x)) = f'(g(x))g'(x)
The individual derivatives are:
  • f'(g) = cos(g)
  • g'(x) = 2x
ddxsin(x2) = cos(g(x)) (2x)
= 2x cos(x2)
Another way of writing the Chain Rule is: dydx = dydududx
Let's do the previous example again using that formula:

Example: What is ddxsin(x2) ?

dydx = dydududx
Have u = x2, so y = sin(u):
ddx sin(x2) = ddusin(u)ddxx2
Differentiate each:
ddx sin(x2) = cos(u) (2x)
Substitue back u = x2 and simplify:
ddx sin(x2) = 2x cos(x2)
Same result as before (thank goodness!)
Another couple of examples of the Chain Rule:

Example: What is d/dx(1/cos(x)) ?

1/cos(x) is made up of 1/g and cos():
  • f(g) = 1/g
  • g(x) = cos(x)
The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
The individual derivatives are:
  • f'(g) = −1/(g2)
  • g'(x) = −sin(x)
(1/cos(x))’ = −1/(g(x))2 × −sin(x)
= sin(x)/cos2(x)
Note: sin(x)/cos2(x) is also tan(x)/cos(x), or many other forms.

Example: What is d/dx(5x−2)3 ?

The Chain Rule says:
the derivative of f(g(x)) = f’(g(x))g’(x)
(5x-2)3 is made up of g3 and 5x-2:
  • f(g) = g3
  • g(x) = 5x−2
The individual derivatives are:
  • f'(g) = 3g2 (by the Power Rule)
  • g'(x) = 5
d/dx(5x−2)3 = 3g(x)2 × 5 = 15(5x−2)2



Admissions,7,Agri-Jobs,1,Animated Moral Stories,2,Atomic-Jobs,1,California Scholarships,1,Canada Scholarships,2,Colleges,1,Computer,32,Conferences,5,CSS Agriculture,3,CSS Current Affairs,4,CSS Data,62,CSS Date Sheet,1,CSS English Essay,10,CSS Everyday Science,7,CSS General Knowledge,1,CSS History of Pakistan,2,CSS Islamiyat,4,CSS Pakistan Affairs,28,CSS Past Paper,59,Czech Scholarships,1,Data,5,Date Sheets,5,Derivative,2,Differentiate between,2,Differentiation,1,Employees,1,English1 FSc Notes,15,English3 FSc Notes,3,English4 FSc Notes,2,Entry Test,1,Essay Writing,3,Exchange Programs,3,Fellowships,4,firdous,1,Food Jobs,2,Form of Verbs,26,Fully Funded Scholarships,16,Funded,1,Germany Scholarships,1,Guess 2nd Year,10,HEC HAT Test,1,Important Questions Chemistry2 FSc,4,Important Questions Physics2 FSc,5,Integration,5,Inter Date Sheet,1,Internships,6,Italy Scholarships,1,Japan Scholarships,2,Jobs,15,Masters Scholarships,16,Moral-Stories,1,NAB-Jobs,1,Netherland Scholarships,1,Netherlands Scholarships,1,News-Updates,6,Novel,1,OP Questions Chemistry2 FSc,4,OP Questions Physics2 FSc,5,Partially Funded,2,PhD Scholarships,8,Presentations,2,Quotes,1,Results,12,Roll Number Slips,1,Scholarships,30,Singapore Scholarships,1,South Korea Scholarships,2,Statistics,6,Switzerland,1,Test Chemistry2 FSc,1,Tips,3,Training Programs,1,Turkey Scholarships,1,Undergraduate Scholarships,7,United Kingdom,2,Urdu Essays,1,USA Scholarships,4,Vacancies,1,Videos,2,Writting Tips,1,
Ilm Gah: Derivative Rules
Derivative Rules
There are rules we can follow to find many derivatives.
Ilm Gah
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED STEP 1: Share to a social network STEP 2: Click the link on your social network Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy