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Thought-Process to Discover Knowledge

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nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

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User Guide

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exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Standard Results in Derivatives

Voice

Voice

Home

»  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)

Progress

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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