# 0.2 Table of derivatives

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## General formulas

1. $\frac{d}{dx}\left(c\right)=0$

2. $\frac{d}{dx}\left(f\left(x\right)+g\left(x\right)\right)={f}^{\prime }\left(x\right)+{g}^{\prime }\left(x\right)$

3. $\frac{d}{dx}\left(f\left(x\right)g\left(x\right)\right)={f}^{\prime }\left(x\right)g\left(x\right)+f\left(x\right){g}^{\prime }\left(x\right)$

4. $\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1},\phantom{\rule{0.2em}{0ex}}\text{for real numbers}\phantom{\rule{0.2em}{0ex}}n$

5. $\frac{d}{dx}\left(cf\left(x\right)\right)=c{f}^{\prime }\left(x\right)$

6. $\frac{d}{dx}\left(f\left(x\right)-g\left(x\right)\right)={f}^{\prime }\left(x\right)-{g}^{\prime }\left(x\right)$

7. $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{g\left(x\right){f}^{\prime }\left(x\right)-f\left(x\right){g}^{\prime }\left(x\right)}{{\left(g\left(x\right)\right)}^{2}}$

8. $\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime }\left(g\left(x\right)\right)·{g}^{\prime }\left(x\right)$

## Trigonometric functions

9. $\frac{d}{dx}\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x$

10. $\frac{d}{dx}\left(\text{tan}x\right)={\text{sec}}^{2}x$

11. $\frac{d}{dx}\left(\text{sec}x\right)=\text{sec}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}x$

12. $\frac{d}{dx}\left(\text{cos}\phantom{\rule{0.1em}{0ex}}x\right)=\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}x$

13. $\frac{d}{dx}\left(\text{cot}x\right)=\text{−}{\text{csc}}^{2}x$

14. $\frac{d}{dx}\left(\text{csc}x\right)=\text{−csc}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cot}\phantom{\rule{0.1em}{0ex}}x$

## Inverse trigonometric functions

15. $\frac{d}{dx}\left({\text{sin}}^{-1}x\right)=\frac{1}{\sqrt{1-{x}^{2}}}$

16. $\frac{d}{dx}\left({\text{tan}}^{-1}x\right)=\frac{1}{1+{x}^{2}}$

17. $\frac{d}{dx}\left({\text{sec}}^{-1}x\right)=\frac{1}{|x|\sqrt{{x}^{2}-1}}$

18. $\frac{d}{dx}\left({\text{cos}}^{-1}x\right)=-\frac{1}{\sqrt{1-{x}^{2}}}$

19. $\frac{d}{dx}\left({\text{cot}}^{-1}x\right)=-\frac{1}{1+{x}^{2}}$

20. $\frac{d}{dx}\left({\text{csc}}^{-1}x\right)=-\frac{1}{|x|\sqrt{{x}^{2}-1}}$

## Exponential and logarithmic functions

21. $\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$

22. $\frac{d}{dx}\left(\text{ln}\phantom{\rule{0.1em}{0ex}}|x|\right)=\frac{1}{x}$

23. $\frac{d}{dx}\left({b}^{x}\right)={b}^{x}\text{ln}\phantom{\rule{0.1em}{0ex}}b$

24. $\frac{d}{dx}\left({\text{log}}_{b}x\right)=\frac{1}{x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}b}$

## Hyperbolic functions

25. $\frac{d}{dx}\left(\text{sinh}\phantom{\rule{0.1em}{0ex}}x\right)=\text{cosh}\phantom{\rule{0.1em}{0ex}}x$

26. $\frac{d}{dx}\left(\text{tanh}\phantom{\rule{0.1em}{0ex}}x\right)={\text{sech}}^{2}\phantom{\rule{0.1em}{0ex}}x$

27. $\frac{d}{dx}\left(\text{sech}\phantom{\rule{0.1em}{0ex}}x\right)=\text{−sech}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{tanh}\phantom{\rule{0.1em}{0ex}}x$

28. $\frac{d}{dx}\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}x\right)=\text{sinh}\phantom{\rule{0.1em}{0ex}}x$

29. $\frac{d}{dx}\left(\text{coth}\phantom{\rule{0.1em}{0ex}}x\right)=\text{−}{\text{csch}}^{2}\phantom{\rule{0.1em}{0ex}}x$

30. $\frac{d}{dx}\left(\text{csch}\phantom{\rule{0.1em}{0ex}}x\right)=\text{−csch}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{coth}\phantom{\rule{0.1em}{0ex}}x$

## Inverse hyperbolic functions

31. $\frac{d}{dx}\left({\text{sinh}}^{-1}x\right)=\frac{1}{\sqrt{{x}^{2}+1}}$

32. $\frac{d}{dx}\left({\text{tanh}}^{-1}x\right)=\frac{1}{1-{x}^{2}}\left(|x|<1\right)$

33. $\frac{d}{dx}\left({\text{sech}}^{-1}x\right)=-\frac{1}{x\sqrt{1-{x}^{2}}}\phantom{\rule{1em}{0ex}}\left(0

34. $\frac{d}{dx}\left({\text{cosh}}^{-1}x\right)=\frac{1}{\sqrt{{x}^{2}-1}}\phantom{\rule{1em}{0ex}}\left(x>1\right)$

35. $\frac{d}{dx}\left({\text{coth}}^{-1}x\right)=\frac{1}{1-{x}^{2}}\phantom{\rule{1em}{0ex}}\left(|x|>1\right)$

36. $\frac{d}{dx}\left({\text{csch}}^{-1}x\right)=-\frac{1}{|x|\sqrt{1+{x}^{2}}}\phantom{\rule{0.2em}{0ex}}\left(x\ne 0\right)$

#### Questions & Answers

why n does not equal -1
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The
What is limit
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
what is implicit of y³+x²y³=5 at (2,1)
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg