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Thought-Process to Discover Knowledge

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nub is the simple explanation of the concept.

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Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step. continue

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exercise provides practice problems to become fluent in the concepts.

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summary of this topic

Standard Results in Derivatives

Standard Results in Derivatives

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 »  Derivatives of Exponents or Logarithmic Functions
    `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

Derivatives of Exponents and Logarithmic Functions

plain and simple summary

nub

plain and simple summary

nub

dummy

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 `



simple steps to build the foundation

trek

simple steps to build the foundation

trek

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


Keep tapping on the content to continue learning.
starting on learning "Derivatives of Exponents and Logarithmic Functions". In this page, derivatives of exponents and logarithmic functions such as e power x, a power x, and natural log of x.

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^(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 = 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`

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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

  `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



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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)`

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)`

What is the derivative of `e^(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)`

your progress details

Progress

About you

Progress

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?
1
2
The answer is "d by dx of e power x = e power x"
Finding the derivative of y = natural log x is given. What does the above prove.
1
2
The answer is "d by dx of natural log x = 1 by x "
Finding the derivative of y = a power x is given. What does the above prove.
1
2
The answer is "d by dx of a power x = a power x, multiplied, natural log a"
Derivatives of Exponents or Logarithmic Functions are listed
Derivatives of Exponents or Logarithmic Functions are given. The number e is defined such that rate of change is proportional to itself. All follow a certain pattern, quickly follow them to derive the result.
what is the derivative of log base 10 x.
1
2
The answer is "1 by x natural log 10"
What is the derivative of e power a x .
1
2
The answer is "a e power a x"

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