Calculus Cheat Sheet (Important Formulas, Equations)

This calculus cheat sheet summarizes all the key formulas, rules, and concepts for quick reference. Keep it handy while studying or solving problems.

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1. Limits (Foundation of Calculus)

Definition: $\lim_{x \to a} f(x)$limx?a?f(x) is the value $f(x)$f(x) approaches as $x$x approaches $a$a.

Key Rules:

1. Limit of a constant: $\lim_{x \to a} c = c$limx?a?c=c
2. Limit of x: $\lim_{x \to a} x = a$limx?a?x=a
3. Sum/Difference: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$limx?a?[f(x)±g(x)]=limx?a?f(x)±limx?a?g(x)
4. Product: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$limx?a?[f(x)?g(x)]=limx?a?f(x)?limx?a?g(x)
5. Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$limx?a?g(x)f(x)?=limg(x)limf(x)?, $g(a) \neq 0$g(a)?=0

Special Cases:

• Indeterminate forms: $0/0, \infty/\infty$0/0,?/? ? use algebraic simplification or L’Hôpital’s Rule

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2. Derivatives (Rates of Change)

Definition: The derivative $f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$f?(x)=limh?0?hf(x+h)?f(x)? measures instantaneous rate of change.

Basic Rules

1. Power Rule: $\frac{d}{dx} [x^n] = n x^{n-1}$dxd?[xn]=nxn?1
2. Constant Rule: $\frac{d}{dx}[c] = 0$dxd?[c]=0
3. Sum/Difference Rule: $(f \pm g)’ = f’ \pm g’$(f±g)?=f?±g?
4. Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$dxd?[c?f(x)]=c?f?(x)

Product and Quotient Rules

• Product: $(f \cdot g)’ = f’g + fg’$(f?g)?=f?g+fg?
• Quotient: $\left(\frac{f}{g}\right)’ = \frac{f’g – fg’}{g^2}$(gf?)?=g2f?g?fg??

Chain Rule (Composite Functions)

• $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$dxd?f(g(x))=f?(g(x))?g?(x)

Derivatives of Common Functions

Function	Derivative
$x^n$xn	$n x^{n-1}$nxn?1
$e^x$ex	$e^x$ex
$a^x$ax	$a^x \ln a$axlna
$\ln x$lnx	$1/x$1/x
$\sin x$sinx	$\cos x$cosx
$\cos x$cosx	$-\sin x$?sinx
$\tan x$tanx	$\sec^2 x$sec2x
$\cot x$cotx	$-\csc^2 x$?csc2x
$\sec x$secx	$\sec x \tan x$secxtanx
$\csc x$cscx	$-\csc x \cot x$?cscxcotx

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3. Applications of Derivatives

• Slope of tangent line: $y – y_1 = f'(x_1)(x – x_1)$y?y1?=f?(x1?)(x?x1?)
• Critical points: Solve $f'(x) = 0$f?(x)=0
• Maxima/Minima: Use $f”(x)$f??(x):
  • $f”(x) > 0$f??(x)>0 ? local minimum
  • $f”(x) < 0$f??(x)<0 ? local maximum
• Concavity:
  • $f”(x) > 0$f??(x)>0 ? concave up
  • $f”(x) < 0$f??(x)<0 ? concave down
• Inflection points: Solve $f”(x) = 0$f??(x)=0

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4. Integrals (Accumulation & Area)

Definition: $\int f(x) dx$?f(x)dx is the antiderivative; $\int_a^b f(x) dx$?ab?f(x)dx is the area under $f(x)$f(x).

Basic Rules

1. $\int x^n dx = \frac{x^{n+1}}{n+1} + C$?xndx=n+1xn+1?+C ($n \neq -1$n?=?1)
2. $\int c dx = cx + C$?cdx=cx+C
3. $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$?[f(x)±g(x)]dx=?f(x)dx±?g(x)dx
4. Constant multiple: $\int c f(x) dx = c \int f(x) dx$?cf(x)dx=c?f(x)dx

Substitution Rule

• $\int f(g(x)) g'(x) dx = \int f(u) du$?f(g(x))g?(x)dx=?f(u)du, where $u = g(x)$u=g(x)

Integration by Parts

• $\int u dv = uv – \int v du$?udv=uv??vdu

Definite Integrals

• $\int_a^b f(x) dx = F(b) – F(a)$?ab?f(x)dx=F(b)?F(a), where $F(x)$F(x) is antiderivative of $f(x)$f(x)

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5. Common Integral Formulas

Function	Integral
$x^n$xn	$\frac{x^{n+1}}{n+1} + C$n+1xn+1?+C
$e^x$ex	$e^x + C$ex+C
$a^x$ax	$\frac{a^x}{\ln a} + C$lnaax?+C
$\sin x$sinx	$-\cos x + C$?cosx+C
$\cos x$cosx	$\sin x + C$sinx+C
$\sec^2 x$sec2x	$\tan x + C$tanx+C
$\csc^2 x$csc2x	$-\cot x + C$?cotx+C
$\sec x \tan x$secxtanx	$\sec x + C$secx+C
$\csc x \cot x$cscxcotx	$-\csc x + C$?cscx+C

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6. Trigonometric Identities for Calculus

• $\sin^2 x + \cos^2 x = 1$sin2x+cos2x=1
• $1 + \tan^2 x = \sec^2 x$1+tan2x=sec2x
• $1 + \cot^2 x = \csc^2 x$1+cot2x=csc2x
• $\sin(2x) = 2 \sin x \cos x$sin(2x)=2sinxcosx
• $\cos(2x) = \cos^2 x – \sin^2 x$cos(2x)=cos2x?sin2x

? Tip: These are essential for derivatives and integrals of trig functions.

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7. Quick Problem-Solving Tips

1. Derivatives: Always check for chain rule, product rule, quotient rule
2. Integrals: Factor out constants first, look for substitution opportunities
3. Visualization: Sketch curves for slope, concavity, and area
4. Units & Real-life context: Helps verify answers in applied problems
5. Check with derivatives: For integrals, take derivative to verify

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8. Mindset & Study Tricks

• Understand first, memorize second
• Teach someone else to reinforce concepts
• Mix practice problems: polynomials, trig, exponentials, word problems
• Use spaced repetition: review formulas every 2–3 days

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? Summary of Essentials

1. Limits ? foundation of derivatives
2. Derivatives ? slopes, rates of change, applications
3. Integrals ? areas, accumulation, real-life problems
4. Tricks: Chain rule, product/quotient, substitution, integration by parts
5. Visualization + Practice ? key to mastering calculus fast

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With this cheat sheet, you can study daily, review quickly, and apply concepts without flipping through textbooks. Pair it with your 30-day plan, and you’ll be able to understand and solve calculus problems confidently.