Calculus

Chain rule

100

f(x)=(x+1)^2 find the slope of the tangent line at x=2

f'(x)=2(x+1), 2(2+1)=6

200

f(x)=sin(3x)

f'(x)= 3cos(3x)

300

f(x)=4cos(3x/2)

f'(x)=-4sin(3x/2) (3/2)=

-6sin(3x/2)

400

𝑓⁡(𝑥) =(6⁢𝑥^2+7⁢𝑥)^4

𝑓′⁡(𝑥)=4⁢(6⁢𝑥^2+7⁢𝑥)^3⁢(12⁢𝑥+7)=

4⁢(12⁢𝑥+7)⁢(6⁢𝑥^2+7⁢𝑥)^3

500

 ℎ⁡(𝑧) =sin⁡(𝑧^6) +sin^6⁡(𝑧)

ℎ′⁡(𝑧)=6⁢𝑧^5⁢ cos⁡(𝑧^6)+ 6⁢sin^5⁡(𝑧)⁢cos⁡(𝑧)

limits

100

Lim x->3 of (3x^2)

27

200

lim x->5 of ([2x^3]/x)

50

300

lim x->2 of ([7x^3]/9x^2)

56/36

400

lim x->pi of sec(X)

-1

500

lim x->4pi/3 of tan^2(x)

3

Implicit differentiation

100

dy/dx(3x^2+y)

=6x+dy/dx

200

dy/dx{(3x^2)*(y^2)}

=(6xy^2)+(3x^2*2ydy/dx)

300

dy/dx{𝑥^3⁢*𝑦^5 +3⁢𝑥 +10}

{(3x^2*y^5)(5y^4 dy/dx*x^3)+3}

400

dy/dx{(x^4)/(y^2)}

{[(4x^3*y^2)-(x^4*2y*dy/dx)]/(y^2)^2 }

500

dy/dx{(siny)/(x^3)}

{[(cosy*dy/dx*x^3)-(3x^2*siny)]/(x^3)^2}

Indefinite Integrals

100

int(3x^2)dx

x^3+C

200

int(sin(x))dx

-cos(x)+C

300

int{5x^2+sin(3x)}dx

(5x^3)/3 -1/3cos(3x)+C

400

1/3  int{(7y^4)}dy

1/15(7y^5)+C

500

int{(x+x^1/3)(4-x^2)}dx

=int{4x-x^3+4x^1/3-x^7/3}

=2x^2   -1/4(x^4)   +3x^4/3   -3/10(x^10/3)+C

Trig Integration

100

avg value of int from 0-2 of (siny)dy

1/2[-cos(2)+1]

200

int(3cos2x)dx

(3/2)sin(2x)+C

300

int([-1/2][cos4x])dx

-1/8sin4x+C

400

int(tanx)dx

-ln(abs val[cosx])+C

500

int[du/(a^2+u^2)]

1/a tan^-1(u/a)+C