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Thought-Process to Discover Knowledge

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mathsDifferential CalculusAlgebra of Differentiation

Understanding Algebra of Derivatives

In this page, what is algebra of derivatives and conditions under which it is applicable are discussed.



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What does the title "Algebra of differentiation" or "Algebra of derivatives" mean?

  • Properties to find derivatives of functions given as algebraic operations of several functions
  • Properties to find derivatives of functions given as algebraic operations of several functions
  • application of differentiation

The answer is "Properties to find derivatives of functions given as algebraic operations of several functions"

The mathematical operations are

 •  addition and subtraction `u(x) +- v(x)`

 •  multiple of a function `a u(x)`

 •  multiplication and division `u(x)v(x)` and `(u(x))/(v(x))`

 •  powers and roots `[u(x)]^n` and `[u(x)]^(1/n)`

 •  composite form of functions `v (u(x))`

 •  parametric form of functions `v=f(r) ; u=g(r)`

Given that `f(x) = u(x)***v(x)` where `***` is one of the arithmetic or function operations.

Will there be any relationship between the derivative of the functions `d/(dx) u(x)` ; `d/(dx) v(x)` and the derivative of the result `d/(dx) f(x)`?

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 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.

For example, consider
`u(x) = x^2`
`v(x) = sin x`
`f(x) = x^2 sin x`

From the standard results, it is known that
`d/(dx) x^2 = 2x` and
`d/(dx) sin x = cos x`.
What is `d/(dx) x^2 sin x`?

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.

                            
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