Studied by 42 peopleY=f(x) must be continuous at each:
-Critical Point or undefined and endpoints
Local Minimum
Goes (-,0,+) or (-, und, +)
Local Maximum
Goes (+,0,-) or (+, und, -)
Point of Inflection
Concavity Changes
(+,0,-) or (-,0,+)
(+,und,-) or (-,und,+)
D/dx(sinx)
Cosx
D/dx(cosx)
-sinx
D/dx(tanx)
Sec²x
D/dx(cotx)
-csc²x
D/dx(secx)
Secxtanx
D/dx(cscx)
-cscxcotx
D/dx (lnx)
1/x × derivative of x
D/dx(eⁿ)
Eⁿ derivative of n
∫Cosx
-Sinx
∫-sinx
Cosx
∫Sec²x
Tanx
∫-csc²x
Cotx
∫Secxtanx
Secx
∫-cscxcotx
Cscx
∫1/n
Ln(n)
∫Eⁿ
Eⁿ
When doing integrals never forget
Constant (+c)
∫Axⁿ
A/n+1(xⁿ⁺¹)+C
∫Tanx
Ln|secx|+c
-Ln|cosx|+c
∫Secx
Ln|secx+tanx|+c
D/dx(sin⁻¹x)
1/√1-x²
D/dx(cos⁻¹x)
-1/√1-x²
D/dx(tan⁻¹x)
1/1+x²
D/dx(cot⁻¹x)
-1/1+x²
With derivative inverses
You plug in the number of the trigonometric function into x
D/dx(sec⁻¹x)
1/|x|√x²-1
D/dx(csc⁻¹x)
-1/|x|√x²-1
D/dx(aⁿ)
aⁿln(a)
D/dx(Logₙx)
1/xln(a)
Chain Rule
Take derivative of outside of parenthesis
Take derivative of inside parenthesis and keep the original of what was in the parenthesis
For example, sin(x²+1)→ 2xcos(x²+1)
Product Rule
d/dx first times second + first times d/dx second
Quotient Rule
LoDHi-HiDLo/LoLo
Fundamental Theorem of Calculus
∫(a to b) f(x) dx = F(b) - F(a)
Basically saying that F’(x)=f(x)
f relative max→f ‘ goes from
Positive to negative
f relative min→f ‘ goes from
Negative to positive
Note
Flashcard