Overview
» Understanding Algebra of Derivatives
how derivative applies to a function given as algebraic operations of several functions
→ addition and subtraction
→ product and division
→ function-of-function
→ parametric form of function
» Differentiation under Basic Arithmetic Operations
→
→
→
→
→
» Differentiation under Function Operations
→ Composite form and Chain rule: given and (i.e. ) then
→ Parametric form : given and then
algebra is about operations + - * / ^
"Algebra of differentiation" or "Algebra of derivatives" means studying of "Properties to find derivatives of functions given as algebraic operations of several functions".
The mathematical operations with functions and are
• addition and subtraction
• multiple of a function
• multiplication and division and
• powers and roots and
• composite form of functions
• parametric form of functions
establishing the problem
Given that where is one of the arithmetic or function operations.
Will there be any relationship between the derivative of the functions ; and the derivative of the result ?
Algebra of differentiation analyses this and provides the required knowledge.
Note: In deriving the results, the functions are assumed to be continuous and differentiable at the points or range of interest. For specific functions at specific values of variables, one must check for the continuity and the differentiability before using the algebra of derivatives.
example problem
For example, consider
From the standard results, it is known that
and
.
What is ?
In this particular example multiplication is considered. Instead of multiplication, one of the arithmetic or function operations may be considered too.
The algebra of derivatives analyses this and provides the required knowledge.
scalar multiple
Derivative of a scalar multiple of function.
Given .
with continuity and differentiability conditions on and
The above proves that . In other words, derivative of a multiple of a function is multiple of the derivative of the function
Intuitive understanding for
rate of change multiplies when the function is multiplied by a constant.
Given and , what is ?
The answer is ""
summary
Derivative of a Multiple:
Derivative of a multiple of a function is multiple of the derivative of the function.
sum or difference
Finding derivative of sum or difference.
with continuity and differentiability conditions on and
The above proves and . In other words, Derivative of a sum or difference is the sum or difference of derivatives.
Intuitive understanding for
rate of change adds (subtracts) when the function is added(subtracted).
Given and , what is ?
The answer is ""
summary
Derivative of Sum or Difference:
Derivative of a sum or difference is the sum or difference of derivatives.
product
Finding derivative of product.
adding
with continuity and differentiability conditions on and
The above proves that
Intuitive understanding of
Two functions are multiplied at every point.
Rate of change of the product equals sum of
fixing one function as a constant and rate of change of the other function
plus fixing the other function as a constant and rate of change of the function
This is compared to .
Given what is ?
Note:
The answer is ""
applying product law of derivative
summary
Derivative of Product:
Derivative of product of two function is sum of derivatives of the functions scaled by the value of other function
division
Finding derivative of division.
adding
with continuity and differentiability conditions on and
The above proves
Intuitive understanding of
This can be rewritten as
The first part of the result is :
fixing as a constant, rate of change of is taken
The second part of the result is :
fixing as a constant, rate of change of is taken.
Rate of change of is negative of rate of change of and divided by .
Given what is ?
Note:
The answer is ""
applying quotient law of derivative
summary
Derivative of Quotient:
composite
Finding derivative of composite function.
Given and find
multiplying and dividing by
Substituting
If , then
with continuity and differentiability conditions on and .
The above proves
Derivative of composite function can be extended to multi-level functions.
This is also called chain rule of differentiation.
Intuitive understanding of chain rule :
In , is called outer-function and is called inner-function.
In a composite function,
the change in variable causes a change in inner-function and that change is the rate of change of inner-function.
the change in inner-function, in turn, causes change in outer-function. This change, rate of change of outer-function, is with respect to the inner-function.
Thus rate of change of outer function with respect to variable rate of change of inner-function to the variable rate of change of outer-function to the inner-function.
Given , what is ?
Note:
The answer is ""
taking
applying law of derivative
Derivative of Composite Function :
parametric form
Finding derivative of function given in parametric form.
and
Note that change in will reflect as change in .
dividing numerator and denominator by
with continuity and differentiability conditions on and
The above proves
Given and , what is ?
Note:
?
The answer is ""
as and
Intuitive understanding of Rate of change of a function with respect to another function can be computed by the change in the common variable . The change in the common variable causes change in as well as in . This leads to the given formula.
summary
Derivative of function in parametric form:
summary
Derivative of a Multiple:
Derivative of a multiple of a function is multiple of the derivative of the function.
Derivative of Sum or Difference:
Derivative of a sum or difference is the sum or difference of derivatives.
Derivative of Product:
Derivative of product of two function is sum of derivatives of the functions scaled by the value of other function
Derivative of Quotient:
Derivative of function in parametric form:
Outline
The outline of material to learn "Differential Calculus" is as follows.
• Detailed outline of Differential Calculus
→ Application Scenario
→ Differentiation in First Principles
→ Graphical Meaning of Differentiation
→ Differntiability
→ Algebra of Derivatives
→ Standard Results