Calculus Formulae

# Differentiation

## Derivatives

• Simple Polynomials: $\frac{d}{dx} x^n = nx^{n-1}$
• Trigonometric Functions: $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$, $\frac{d}{dx} \tan x = \sec ^2 x$
• Exponential Functions: For $a > 0$, $\frac{d}{dx} a^x = a^x \ln a$, and $\frac{d}{dx} e^x = e^x$
• Logarithmic Functions: $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$
• Inverse Trigonometric Functions: $\frac{d}{dx} \sin ^{-1} x = \frac{1}{\sqrt{1 - x^2}}$, $\frac{d}{dx} \cos ^{-1} x = - \frac{1}{\sqrt{1 - x^2}}$, $\frac{d}{dx} \tan ^{-1} x = \frac{1}{1 + x^2}$
• Reciprocal Trigonometric Functions: $\frac{d}{dx} \sec x = \sec x \tan x$, $\frac{d}{dx} \csc x = -\csc x \cot x$, $\frac{d}{dx} \cot x = -\csc ^2 x$.
• Inverse Reciprocal Trigonometric Functions: $\frac{d}{dx} \sec ^{-1} x = \frac{1}{\mid x \mid \sqrt{x^2 - 1}}$, $\frac{d}{dx} \csc ^{-1} x = - \frac{1}{\mid x \mid \sqrt{x^2 - 1}}$, $\cot ^{-1} x = -\frac{1}{1 + x^2}$

## Differentiation Rules

• Product Rule: $\frac{d}{dx} fg = fg' + f'g$
• Quotient Rule: $\frac{d}{dx} f/g = \frac{gf' - fg'}{g^2}$, $g ≠ 0$
• Chain Rule: $\frac{d}{dx} f \circ g = f'(g(x))g'(x)$

# Integration

## Integrals

• Simple Polynomials: $\int bx^n \: dx = \frac{bx^{n+1}}{n+1} + C$ for $n ≠ -1$, $\int \frac{1}{x} \: dx = \ln \mid x \mid$
• Trigonometric Functions: $\int \sin x \: dx = -\cos x + C$, $\int \cos x \: dx = \sin x + C$, $\int \tan x \: dx = -\ln \mid \cos x \mid + C = \ln \mid \sec x \mid + C$
• Exponential Functions: $\int a^x \: dx = \frac{a^x}{\ln a} + C$, $\int e^x \: dx = e^x + C$
• Logarithmic Functions: $\int \log_a x \: dx = \frac{x \log x - x}{\log a} + C$, $\int \ln x = x \log x - x$
• Squared Trigonometric Functions: $\int \sin ^2 x \: dx = \frac{x - \sin x \cos x}{2} + C$, $\int \cos ^2 x \: dx = \frac{\sin x \cos x + x}{2} + C$, $\int \tan ^2 x \: dx = \tan x - x + C$
• Reciprocal Trigonometric Functions: $\int \sec x \: dx = \ln \mid \sec x + \tan x \mid + C$, $\int \csc \: dx = - \ln \mid \csc x + \cot x \mid + C$, $\int \cot x \: dx = \ln \mid \sin x \mid + C$
• Inverse Trigonometric Functions: $\int \sin ^{-1} x \: dx = x \sin ^{-1} x + \sqrt{1 - x^2} + C$, $\int \cos ^{-1} x \: dx = x \cos ^{-1} x - \sqrt{1 - x^2} + C$, $\int \tan ^{-1} x \: dx = x \tan ^{-1} x - \frac{1}{2} \ln (1 + x^2) + C$

## Integration Rules

• Definite Integrals: $\int_a^b f(x) \: dx = F(x) \: \biggr\rvert_{a}^{b} = F(b) - F(a)$
• Integration by Parts: $\int u \: dv = uv - \int v \: du$

# Area Formulae

• Area Trapped Between a Function and the X-axis: For $y = f(x) ≥ 0$ and $a < b$, $A = \int_a^b f(x) \: dx$.
• Area Trapped Between a Curve and the Y-Axis: For $x = f(y) ≥ 0$ and $c < d$, $A = \int_c^d f(y) \: dy$.
• Area Trapped Between Two Curves: For $f(x) ≥ g(x) ≥ 0$ on the interval $[a, b]$, $A = \int_a^b [f(x) - g(x)] \: dx$.
• Area of a Parametric Curve: If $x = f(t)$ and $y = g(t)$, $A = \int_{\alpha}^{\beta} g(t) f'(t) \: dt$.
• Area of a Polar Curve: If $r = f(\theta)$, $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \: d \theta$.

# Volume Formulae

• Washer/Disk Method: $V = \pi \int_a^b [f(x)]^2 \: dx$
• Cylindrical Shells Method: $V = 2\pi \int_a^b xf(x) \: dx$

# Limit Proof Implications

• Existence of a Regular Limit: $\lim_{x \to a} f(x) = L \Leftrightarrow \forall \epsilon > 0 \: \exists \delta > 0 \: \mathrm{s.t.} \: \forall x : 0 < \mid x - a \mid < \delta, \: \mid f(x) - L \mid < \epsilon$.
• Nonexistence of a Regular Limit: $\lim_{x \to a} f(x) ≠ L \Leftrightarrow \exists \epsilon > 0 \: \forall \delta > 0 \: \mathrm{s.t.} \: \exists x : 0 < \mid x - a \mid < \delta, \: \mid f(x) - L \mid ≥ \epsilon$.
• Limit to Positive Infinity: $\lim_{x \to a} f(x) = \infty \Leftrightarrow \forall m > 0, \exists \delta > 0 \: \mathrm{s.t.} \: \forall x : 0 < \mid x - a \mid < \delta, \: f(x) > m$.
• Limit to Negative Infinity: $\lim_{x \to a} f(x) = -\infty \Leftrightarrow \forall m < 0, \exists \delta > 0 \: \mathrm{s.t.} \: \forall x : 0 < \mid x - a \mid < \delta, \: f(x) < m$.
• Limit at Infinity: $\lim_{x \to \infty} f(x) = L \Leftrightarrow \forall \epsilon > 0 \: \exists k \in \mathbb{R} \: \mathrm{s.t.} \: \forall x : x > k, \: \mid f(x) - L \mid < \epsilon$.
• Limit at Negative Infinity: $\lim_{x \to -\infty} f(x) = L \Leftrightarrow \forall \epsilon > 0 \: \exists k \in \mathbb{R} \: \mathrm{s.t.} \: \forall x : x < k \: \mid f(x) - L \mid < \epsilon$.
• Limit to Infinity at Infinity: $\lim_{x \to \infty} f(x) = \infty \Leftrightarrow \forall m > 0 \: \exists k \in \mathbb{R} \: \mathrm{s.t.} \: \forall x : x > k, \: f(x) > m$.
• Limit to Negative Infinity at Negative Infinity: $\lim_{x \to -\infty} f(x) = -\infty \Leftrightarrow \forall m < 0 \: \exists k \in \mathbb{R} \: \mathrm{s.t.} \: \forall x : x < k, \: f(x) < m$.