Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsDifferential CalculusStandard Results in Derivatives

Derivatives of Algebraic Expressions

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

click on the content to continue..

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


`=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:


`=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`"

slide-show version coming soon