Chain rule generalized formulas

[Eigenvalue]

Here is a list of functions that will be included in the generalized formulas:

f(x)=x
f(x)=x²
f(x)=x³
f(x)=[tex]\frac{1}{x}[/tex]
f(x)=[tex]\frac{1}{x²}[/tex]
f(x)=[tex]\sqrt{x}[/tex]
f(x)=[tex]\sqrt[3]{x}[/tex]
f(x)=|x|
f(x)=sin(x)
f(x)=cos(x)
f(x)=tan(x)
f(x)=csc(x)
f(x)=sec(x)
f(x)=cot(x)
f(x)=sinh(x)
f(x)=cosh(x)
f(x)=tanh(x)
f(x)=csch(x)
f(x)=sech(x)
f(x)=coth(x)
f(x)=versin(x)
f(x)=vercos(x)

[Eigenvalue]

Thought process:

There are 484 possible generalized formulas as there are 22 possible choices for the first function and 22 possible choices for the second function. It is not needed to divide by 2, as the derivative of f(g(x)) differs from that of g(f(x))

[Eigenvalue]

1. y=u
u=x
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=x'=1
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*1=1

2. y=u
u=x²
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(x²)'=2[tex]x^{(2-1) }[/tex]=2x
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*2x=2x

3. y=u
u=x³
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(x³)'=3[tex]x^{(3-1) }[/tex]=3x²
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*3x²=3x²

4. y=u
u=[tex]\frac{1}{x}[/tex]=[tex]x^{-1}[/tex]
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=-[tex]x^{-1-1 }[/tex]=-[tex]x^{-2 }[/tex]
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*(-[tex]x^{-2 }[/tex])=[tex]\frac{-1}{x²}[/tex]

4. y=u
u=[tex]\frac{1}{x²}[/tex]=[tex]x^{-2 }[/tex]
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=-2[tex]x^{-2-1 }[/tex]=-2[tex]x^{-3 }[/tex]
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*(-2[tex]x^{-3 }[/tex])=[tex]\frac{-2}{x³}[/tex]

[Eigenvalue]

5. y=u
u=sin(x)
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(sin(x))'=cos(x)
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*cos(x)=cos(x)

6. y=u
u=cos(x)
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(cos(x))'=-sin(x)
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*(-sin(x))=-sin(x)

7. y=u
u=tan(x)
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(tan(x))'=sec²(x)
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*sec²(x)=sec²(x)

8. y=u
u=csc(x)
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(csc(x))'=-csc(x)cot(x)
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*(-csc(x)cot(x))=-csc(x)cot(x)

9. y=u
u=sec(x)
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(sec(x))'=sec(x)tan(x)
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*(sec(x)tan(x))=sec(x)tan(x)

10. y=u
u=cot(x)
a) [tex]\frac{dy}{dx}[/tex]=[tex]\frac{dy}{du}[/tex]*[tex]\frac{du}{dx}[/tex]
b)Find [tex]\frac{dy}{du}[/tex]
c)[tex]\frac{dy}{du}[/tex]=u'=1
d) Find [tex]\frac{du}{dx}[/tex]
[tex]\frac{du}{dx}[/tex]=(cot(x))'=-csc²(x)
e) Find the product of [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex]
[tex]\frac{dy}{dx}[/tex]=1*(-csc²(x))=-csc²(x)