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Thought-Process to Discover Knowledge

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

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mathsDifferential CalculusStandard Results in Derivatives

Derivatives of Exponents and Logarithmic Functions

In this page, derivatives of exponents and logarithmic functions such as `e^x`, `a^x`, and `ln x`



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Finding the derivative of `y=e^(x)` in first principles:

`d/(dx) e^(x)`

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

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

`=lim_(delta->0) e^x[color(coral)( e^(delta))``- color(deepskyblue)(1)]//delta`

applying the standard limit `lim_(p->0)(e^p - 1)//p = 1`

`= e^(x)`

What does the above prove?

  • `d/(dx) e^(x) = e^(x)`
  • `d/(dx) e^(x) = e^(x)`
  • `d/(dx) e^(ax) = e^(ax)`

The answer is "`d/(dx) e^(x) = e^(x)`".

Finding the derivative of `y=ln x` :

`y=ln x`

`e^y=x`

differentiating the equation
`(d)/(dx)e^y=1`

applying chain rule `(d)/(dy)e^y (dy)/(dx)=1`

`e^y (dy)/(dx)=1`

`x (dy)/(dx)=1`

`(dy)/(dx)=1/x`

What does the above prove?

  • `d/(dx) ln x = 1/x`
  • `d/(dx) ln x = 1/x`
  • `d/(dx) ln x = cos x + sin x`

The answer is "`d/(dx) ln x = 1/x`"

Finding the derivative of `y=a^x` :

substituting `a=e^(ln a)`

`y=e^(x ln a )`

differentiating the equation
`(dy)/(dx)=(d)/(dx) e^(x ln a )`

applying chain rule with `u=x ln a`
`(dy)/(dx)=(d)/(du) e^u (d)/(dx)(x ln a )`

`(dy)/(dx)= e^u xx ln a`

`(dy)/(dx)= e^(x ln a ) xx ln a `

substituting `e^(ln a)=a`

`(dy)/(dx)=a^x ln a `

What does the above prove?

  • `d/(dx) a^x = a^x ln a`
  • `d/(dx) a^x = a^x ln a`
  • `d/(dx) a^x = a^x`

The answer is "`d/(dx) a^x = a^x ln a`"

Derivatives of Exponents or Logarithmic Functions:
`d/(dx) e^x = e^x`

`d/(dx) a^x = a^x ln a`

`d/(dx) ln x = 1/x `



  `d/(dx) e^x = e^x`
definition of `e` is rate of change is proportional to itself

  `d/(dx) a^x = a^x ln a`
`a` equals `e^(lna)`

  `d/(dx) ln x = 1/x `
natural log is inverse of `e` power

Solved Exercise Problem:

What is the derivative of `log_(10) x`?
Note: Use the identity `log_(10) x = (log_e x)/(log_e 10)`

  • `1/(10x)`
  • `1/(xln 10)`
  • `1/(xln 10)`

The answer is "`1/(xln 10)`".
` d/(dx)log_10 x`
`= d/(dx)(log_e x)/(log_e 10)`
`= 1/(log_e 10) d/(dx) log_e x`
`=1/(log_e 10) xx 1/x `
`=1/(xlog_e 10)`

Solved Exercise Problem:

What is the derivative of `e^(ax)`?

  • `ae^(ax)`
  • `ae^(ax)`
  • `e^(ax)`

The answer is "`ae^(ax)`". Applying chain rule with `u=ax`,
`d/(du) e^u d/(dx) ax`
`e^u xx a`
`ae^(ax)`

                            
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