INTRODUCTION:
A derivative is a method of finding a function derivative at a
given point, and an derivative is an Instantaneous Rate in the
function for one of its variables.
We assume that you have a mathematical equation with a
variable x and y, so in this case, the real-time change rate that
we have just talked about, is a way in which you know how fast
y changes for x at any given value of x, that is, the derivative
applies to it, and that the derivative is a way to find this rate that
we talked about.
DERIVATIVE RULES IN MATHEMATICS
Before talking about the various laws that fall under the
derivative, you should know an important piece of information:
If y is a function of a variable x (in other words: The y equals
the equation of the variable in which it is x, such a relationship:
y = 2x + 1), it means that the y-derivative is equal to
𝑑𝑦
𝑑𝑥
,a
formula that expresses a y-change rate relative to x.
EXPONENTIAL FUNCTION DERIVATIVE
If y is a function of an x variable, and x here is an S variable
(topped by a number, such as x2), then this equation is an
exponential equation, and has a certain method of derivation:
1. If the equation to be derived is: 𝒚 = 𝒙𝒏 , the y derivative
we express with
𝒅𝒚
𝒅𝒙
is equal to: 𝒏𝒙𝒏−𝟏 .
2. If the equation to be derived is: 𝒚 = 𝒌𝒙𝒏 , then:
𝒅𝒚
𝒅𝒙
=
𝒏𝒌𝒙𝒏−𝟏
Which means that the derivative of the exponential function is
to lower the exponent in front of the variable (multiplied by),
then subtract from the exponent one, as we saw in the previous
paragraph.
Examples of Exponential function derivative
 If ( 𝒚 = 𝒙𝟖 ) then (
𝒅𝒚
𝒅𝒙
 If ( 𝒚 = 𝟑𝒙𝟒 ) then (
= 𝟖𝒙𝟕 )
𝒅𝒚
𝒅𝒙
= 𝟏𝟐𝒙𝟑 )
DERIVATIVE OF SUMMED OR
SUBTRACTED FUNCTIONS
The variables in the equation can be a combination or a
subtraction, here, we will derive each variable from the
variables individually, keeping the combination and subtraction
marks in place.
Examples of derivative of summed or subtracted
functions
 If ( 𝒚 = 𝟐𝒙 + 𝟒𝒙𝟑 ) then (
𝒅𝒚
 If ( 𝒚 = 𝒙𝟒 − 𝟑𝒙𝟔 ) then (
𝒅𝒚
𝒅𝒙
𝒅𝒙
= 𝟐 + 𝟏𝟐𝒙𝟐 )
= 𝟒𝒙𝟑 − 𝟏𝟖𝒙𝟓 )
NOTE : We saw in the previous example that the derivative
of 3x is equal to 3, and that's because the x variable here is
considered to has a single exponent, so when we multiply one
in three in front of the variable, the result is 3, and we
subtract one in one which is zero, and there is a rule says that
any value raised to the exponent 0 is equal to 1, so the result
of the derivation of 3x is 3.
FRACTIONAL FUNCTIONS DERIVATIVE
These functions are one of the forms most students make
difficult, because of their complex formulas, but here the
simplicity is the master of the situation, so if the formula is
simplified, the derivation will be easy. The fractional function is
related to the exponential function, where the idea of derivation
of the fractional function lies in its transformation into an
exponential image, and through it is derived according to the
laws of the exponential function.
And there is also a rule we can depends on in fractional
functions derivative.
For example: if ( 𝒚 =
𝒂
𝒅𝒚
𝒙
𝒅𝒙
𝒏 ) then: (
=
𝒂∗(−𝒏)
𝒙𝒏+𝟏
)
Examples of fractional function derivative
𝟏
𝒅𝒚
𝒙
𝒅𝒙
 If ( 𝒚 = ) then: (
 If ( 𝒚 =
−𝟐
𝟑𝒙+𝟏
=
𝒅𝒚
) then: (
𝒅𝒙
−𝟏
𝒙𝟐
=
)
𝟐
)
(𝟑𝒙+𝟏)𝟐 +𝟑
NOTE: we shouldn’t forget that the 3x has to be derived too ,
and that is where +3 come from.
DERIVATIVE OF THE MULTIPLIED
FUNCTION
When I have two functions multiplied in one equation, the
derivative of this equation is equal to: the sum of each: the first
function is multiplied by the derivative of the second function
and the second function is multiplied by the derivative of the
first function.
𝒅𝒚
For example: if ( 𝒚 = (𝒙𝒏 )(𝟏 + 𝒙𝒎 )) then: (
𝒅𝒙
=
𝒙𝒏 (𝒎𝒙𝒎−𝟏 ) + 𝒏𝒙𝒏−𝟏 (𝟏 + 𝒙𝒎 ) )
Examples of Derivative of the multiplied function
 if ( 𝒚 = (𝒙𝟐 )(𝟑𝒙 + 𝟏))
then: (
𝒅𝒚
𝒅𝒙
= 𝒙𝟐 (𝟑) + 𝟐𝒙(𝟑𝒙 + 𝟏) )
CONCLUSION
By the end of reading this report you should have learnt how
to derivate an Exponential, summed or subtracted , Fractional
and Multiplied functions .
REFERENCES
1. Finding Instantaneous Rate of Change of a Function:
Formula & Examples, from the website:
www.study.com
2. Differentiation, from the website:
www.revisionmaths.com
3. Derivative Rules, from the website:
www.mathsisfun.com
4. Derivative of the product of two functions, from the
website: www.sangakoo.com