Calculus Essentials
Derivatives, Integrals &
Applications
Your premium, all-in-one exam-prep study guide 4 built for students who want to ace every calculus exam. Master
derivatives, integrals, and real-world applications with high-yield formulas, memory mnemonics, step-by-step worked
examples, and 30+ practice problems with full solutions.
{ PREMIUM STUDY GUIDE EXAM-READY Ì HIGH-YIELD
, x Overview & Key Formulas
Calculus is the mathematics of change (derivatives) and accumulation (integrals). Master these core concepts and you
unlock physics, engineering, economics, and beyond. The Fundamental Theorem of Calculus unifies both:
+a f 2 (x) dx = f (b) 2 f (a).
b
í Derivative Mnemonic 4 "PQ QP": Quotient Rule = P·Q' minus Q·P', all over Q². Product Rule 4 "First
Derive Second + Second Derive First": (fg)' = f·g' + g·f'. Chain Rule 4 "Outside × Inside": Differentiate the
outer function, keep the inner, multiply by the inner's derivative.
Ì High-Yield Derivative Rules
These appear on nearly every calculus exam 4 memorize first!
Rule Formula Memory Hook
Power Rule ' dx (x )
d n
= nxn21 "Bring down, subtract one"
Product Rule ' (f g)2 = f 2 g + f g 2 "First d-Second + Second d-
First"
2
Quotient Rule ' ( fg ) = f 2 g2f g 2
g2
"PQ minus QP over Q-squared"
Chain Rule ' dx f (g(x))
d
= f 2 (g(x)) ç g 2 (x) "Outside × Inside derivative"
Sine d
dx
sin x = cos x Sine ³ Cosine (no sign
change)
Cosine d
dx cos x = 2 sin x Cosine ³ 2Sine (negative!)
Exponential ' d x
dx e = ex "e-to-x is its own derivative"
Natural Log d
dx ln x = 1
x "Log becomes reciprocal"
Ì High-Yield Integral Rules
Integration is the reverse of differentiation. Always add +C for indefinite integrals!
Rule Formula Memory Hook
Power Rule ' + xn dx = xn+1
n+1
+ C, n =
à 21 "Add one, divide by new
power"
Log Rule ' + 1
x dx = ln #x# + C "Power rule fails at n=21 ³ ln"
Sine + sin x dx = 2 cos x + C "+sin = 2cos (negative!)"
Cosine + cos x dx = sin x + C "+cos = sin (positive)"
Exponential ' + ex dx = ex + C "e-to-x integrates to itself"
Fundamental Theorem ' "Evaluate antiderivative at
b
+a f (x) dx = F (b) 2 F (a)
bounds"
¦ Common Exam Traps: (1) Forgetting +C on indefinite integrals. (2) Missing the negative in
+ sin x dx = 2 cos x + C . (3) Using power rule for 1/x instead of ln rule. (4) Forgetting chain rule on composite
functions.
Derivatives, Integrals &
Applications
Your premium, all-in-one exam-prep study guide 4 built for students who want to ace every calculus exam. Master
derivatives, integrals, and real-world applications with high-yield formulas, memory mnemonics, step-by-step worked
examples, and 30+ practice problems with full solutions.
{ PREMIUM STUDY GUIDE EXAM-READY Ì HIGH-YIELD
, x Overview & Key Formulas
Calculus is the mathematics of change (derivatives) and accumulation (integrals). Master these core concepts and you
unlock physics, engineering, economics, and beyond. The Fundamental Theorem of Calculus unifies both:
+a f 2 (x) dx = f (b) 2 f (a).
b
í Derivative Mnemonic 4 "PQ QP": Quotient Rule = P·Q' minus Q·P', all over Q². Product Rule 4 "First
Derive Second + Second Derive First": (fg)' = f·g' + g·f'. Chain Rule 4 "Outside × Inside": Differentiate the
outer function, keep the inner, multiply by the inner's derivative.
Ì High-Yield Derivative Rules
These appear on nearly every calculus exam 4 memorize first!
Rule Formula Memory Hook
Power Rule ' dx (x )
d n
= nxn21 "Bring down, subtract one"
Product Rule ' (f g)2 = f 2 g + f g 2 "First d-Second + Second d-
First"
2
Quotient Rule ' ( fg ) = f 2 g2f g 2
g2
"PQ minus QP over Q-squared"
Chain Rule ' dx f (g(x))
d
= f 2 (g(x)) ç g 2 (x) "Outside × Inside derivative"
Sine d
dx
sin x = cos x Sine ³ Cosine (no sign
change)
Cosine d
dx cos x = 2 sin x Cosine ³ 2Sine (negative!)
Exponential ' d x
dx e = ex "e-to-x is its own derivative"
Natural Log d
dx ln x = 1
x "Log becomes reciprocal"
Ì High-Yield Integral Rules
Integration is the reverse of differentiation. Always add +C for indefinite integrals!
Rule Formula Memory Hook
Power Rule ' + xn dx = xn+1
n+1
+ C, n =
à 21 "Add one, divide by new
power"
Log Rule ' + 1
x dx = ln #x# + C "Power rule fails at n=21 ³ ln"
Sine + sin x dx = 2 cos x + C "+sin = 2cos (negative!)"
Cosine + cos x dx = sin x + C "+cos = sin (positive)"
Exponential ' + ex dx = ex + C "e-to-x integrates to itself"
Fundamental Theorem ' "Evaluate antiderivative at
b
+a f (x) dx = F (b) 2 F (a)
bounds"
¦ Common Exam Traps: (1) Forgetting +C on indefinite integrals. (2) Missing the negative in
+ sin x dx = 2 cos x + C . (3) Using power rule for 1/x instead of ln rule. (4) Forgetting chain rule on composite
functions.
