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mathsDifferential CalculusStandard Results in Derivatives

Derivatives of Algebraic Expressions

In this page, Derivatives of standard functions in algebraic expressions are explained.



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Finding the derivative of `y=a` in first principles:

`(dy)/(dx)`

`=lim_(delta->0) [color(coral)(f(x+delta))``- color(deepskyblue)(f(x))]//delta`

`=lim_(delta->0) [color(coral)(a)`` - color(deepskyblue)(a)]//delta`

`=lim_(delta->0) [0]//delta`

What does the above prove?

  • `d/(dx) (a) = 0//0`: indeterminate value
  • `d/(dx) (a) = 0`
  • `d/(dx) (a) = 0`

The answer is "`d/(dx) (a) = 0`". Note that `delta` is very small value and the numerator is `0`. So the limit evaluates to `0`.

Finding the derivative of `y=x^n` in first principles:

`(dy)/(dx)`

`=lim_(delta->0) [color(coral)(f(x+delta))``- color(deepskyblue)(f(x))]//delta`

`=lim_(delta->0) [color(coral)((x+delta)^n)`` - color(deepskyblue)(x^n)]//delta`

using binomial expression for `(x+delta)^n` `=lim_(delta->0) [color(coral)((x^n +ndelta x^(n-1)``+ ^nC_2 delta^2x^(n-2)``+ cdots+delta^n)`` - color(deepskyblue)(x^n)]//delta`

canceling `color(coral)(x^n)` and `color(deepskyblue)(-x^n)`
`=lim_(delta->0) [color(coral)((ndelta x^(n-1)``+ ^nC_2 delta^2x^(n-2)``+ cdots+delta^n)``]//delta`

dividing all terms by `delta`
`=lim_(delta->0) [color(coral)((nx^(n-1)``+ ^nC_2 delta^1x^(n-2)``+ cdots+delta^(n-1))``]`

applying limit `delta->0`
What does the above prove?

  • `d/(dx) (x^n) = nx^(n-1)`
  • `d/(dx) (x^n) = nx^(n-1)`
  • `d/(dx) (x^n) = nx^(n)`

The answer is "`d/(dx) (x^n) = nx^(n-1)`". The `delta` is substituted as `0` and only one term is non zero.

`d/(dx) a = 0`

`d/(dx) x^n = n x^(n-1) `

Derivatives of Algebraic Expressions :
  Derivative of a constant
   `d/(dx) a = 0`
rate of change of constant is 0

  Derivative of power:
   `d/(dx) x^n = n x^(n-1) `
small change results in binomial expansion and only one factor remains

Solved Exercise Problem:

Find the derivative of `x^3+4x^2-3x-2`

  • `3x^2+8x-3-2`
  • `3x^2+8x-3`
  • `3x^2+8x-3`
  • `3x^3+4x^2-3x-2`

The answer is "`3x^2+8x-3`"

                            
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