📌 Limits

Limit Definition:

\( \lim_{{x \to a}} f(x) = L \)

Limit of a Constant:

\( \lim_{{x \to a}} c = c \)

Sum Rule:

\( \lim_{{x \to a}} [f(x) + g(x)] = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x) \)

Product Rule:

\( \lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x) \)

Quotient Rule:

\( \lim_{{x \to a}} \frac{{f(x)}}{{g(x)}} = \frac{{\lim_{{x \to a}} f(x)}}{{\lim_{{x \to a}} g(x)}} \), if \( \lim_{{x \to a}} g(x) \neq 0 \)

📌 Basic Differentiation

Derivative of a Constant:

\( \frac{{d}}{{dx}} (c) = 0 \)

Derivative of \( x \):

\( \frac{{d}}{{dx}} (x) = 1 \)

Power Rule (Simple):

\( \frac{{d}}{{dx}} (x^2) = 2x \)

Derivative of Exponential (Base e):

\( \frac{{d}}{{dx}} (e^x) = e^x \)

Derivative of Sine:

\( \frac{{d}}{{dx}} (\sin x) = \cos x \)

📌 Derivative Rules

Definition of Derivative:

\( f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{{h}} \)

General Power Rule:

\( \frac{{d}}{{dx}} (x^n) = n x^{{n-1}} \)

Product Rule:

\( \frac{{d}}{{dx}} [f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \)

Quotient Rule:

\( \frac{{d}}{{dx}} \left( \frac{{f(x)}}{{g(x)}} \right) = \frac{{f'(x)g(x) - f(x)g'(x)}}{{[g(x)]^2}} \)

Chain Rule:

\( \frac{{d}}{{dx}} [f(g(x))] = f'(g(x)) \cdot g'(x) \)

Derivative of Natural Log:

\( \frac{{d}}{{dx}} (\ln x) = \frac{1}{x} \)

Derivative of Tangent:

\( \frac{{d}}{{dx}} (\tan x) = \sec^2 x \)

📌 Implicit Differentiation

Implicit Derivative:

\( \frac{{dy}}{{dx}} = -\frac{{\frac{{\partial F}}{{\partial x}}}}{{\frac{{\partial F}}{{\partial y}}}} \), for \( F(x, y) = 0 \)

Example (\( x^2 + y^2 = 1 \)):

\( 2x + 2y \frac{{dy}}{{dx}} = 0 \Rightarrow \frac{{dy}}{{dx}} = -\frac{{x}}{{y}} \)

📌 Applications of Derivatives

Velocity (First Derivative):

\( v(t) = \frac{{ds}}{{dt}} \)

Acceleration (Second Derivative):

\( a(t) = \frac{{dv}}{{dt}} = \frac{{d^2 s}}{{dt^2}} \)

Critical Points:

\( f'(x) = 0 \) or undefined

Second Derivative Test:

\( f''(x) > 0 \) (min), \( f''(x) < 0 \) (max)

📌 Integrals

Indefinite Integral:

\( \int f(x) \, dx = F(x) + C \), where \( F'(x) = f(x) \)

Power Rule:

\( \int x^n \, dx = \frac{{x^{{n+1}}}}{{n+1}} + C \), \( n \neq -1 \)

Exponential Integral:

\( \int e^x \, dx = e^x + C \)

Trigonometric Integrals:

\( \int \sin x \, dx = -\cos x + C \)

\( \int \cos x \, dx = \sin x + C \)

Definite Integral:

\( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \)

Integration by Parts:

\( \int u \, dv = uv - \int v \, du \)

📌 Applications of Integrals

Area Under Curve:

\( A = \int_{a}^{b} f(x) \, dx \)

Area Between Curves:

\( A = \int_{a}^{b} [f(x) - g(x)] \, dx \), where \( f(x) \geq g(x) \)

Volume of Revolution (Disk Method):

\( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \)

Arc Length:

\( L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx \)

📌 Parametric Equations

Derivative:

\( \frac{{dy}}{{dx}} = \frac{{dy/dt}}{{dx/dt}} \), if \( \frac{{dx}}{{dt}} \neq 0 \)

Arc Length:

\( L = \int_{t_1}^{t_2} \sqrt{\left(\frac{{dx}}{{dt}}\right)^2 + \left(\frac{{dy}}{{dt}}\right)^2} \, dt \)

📌 Polar Coordinates

Conversion to Cartesian:

\( x = r \cos \theta, \, y = r \sin \theta \)

Area in Polar Form:

\( A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 \, d\theta \)

Arc Length:

\( L = \int_{\alpha}^{\beta} \sqrt{[r(\theta)]^2 + \left(\frac{{dr}}{{d\theta}}\right)^2} \, d\theta \)

📌 Improper Integrals

Infinite Limits:

\( \int_{a}^{\infty} f(x) \, dx = \lim_{{t \to \infty}} \int_{a}^{t} f(x) \, dx \)

Discontinuity at \( b \):

\( \int_{a}^{b} f(x) \, dx = \lim_{{t \to b^-}} \int_{a}^{t} f(x) \, dx \)

📌 Differential Equations

Separable Equations:

\( \frac{{dy}}{{dx}} = f(x)g(y) \Rightarrow \int \frac{{dy}}{{g(y)}} = \int f(x) \, dx \)

Linear First-Order:

\( \frac{{dy}}{{dx}} + P(x)y = Q(x) \), solution: \( y = e^{-\int P(x) \, dx} \left( \int Q(x) e^{\int P(x) \, dx} \, dx + C \right) \)

Second-Order (Homogeneous):

\( y'' + py' + qy = 0 \), roots \( r_1, r_2 \): \( y = C_1 e^{r_1 x} + C_2 e^{r_2 x} \)

📌 Series

Geometric Series:

\( \sum_{{n=0}}^{\infty} ar^n = \frac{a}{{1-r}} \), if \( |r| < 1 \)

Taylor Series:

\( f(x) = \sum_{{n=0}}^{\infty} \frac{{f^{(n)}(a)}}{{n!}} (x - a)^n \)

Maclaurin Series for \( e^x \):

\( e^x = \sum_{{n=0}}^{\infty} \frac{{x^n}}{{n!}} \)

Convergence Test (Ratio):

\( \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right| < 1 \) (converges)

📌 Multivariable Calculus

Partial Derivative:

\( \frac{{\partial f}}{{\partial x}} = \lim_{{h \to 0}} \frac{{f(x + h, y) - f(x, y)}}{{h}} \)

Gradient:

\( \nabla f = \left( \frac{{\partial f}}{{\partial x}}, \frac{{\partial f}}{{\partial y}} \right) \)

Double Integral:

\( \iint_{R} f(x, y) \, dA \)

Directional Derivative:

\( D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} \)

📌 Vector Calculus

Divergence:

\( \nabla \cdot \mathbf{F} = \frac{{\partial F_x}}{{\partial x}} + \frac{{\partial F_y}}{{\partial y}} + \frac{{\partial F_z}}{{\partial z}} \)

Curl:

\( \nabla \times \mathbf{F} = \left( \frac{{\partial F_z}}{{\partial y}} - \frac{{\partial F_y}}{{\partial z}}, \frac{{\partial F_x}}{{\partial z}} - \frac{{\partial F_z}}{{\partial x}}, \frac{{\partial F_y}}{{\partial x}} - \frac{{\partial F_x}}{{\partial y}} \right) \)

Green’s Theorem:

\( \oint_{C} (P \, dx + Q \, dy) = \iint_{R} \left( \frac{{\partial Q}}{{\partial x}} - \frac{{\partial P}}{{\partial y}} \right) \, dA \)

Stokes’ Theorem:

\( \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \)

📌 Partial Differential Equations

Wave Equation:

\( \frac{{\partial^2 u}}{{\partial t^2}} = c^2 \frac{{\partial^2 u}}{{\partial x^2}} \)

Heat Equation:

\( \frac{{\partial u}}{{\partial t}} = k \frac{{\partial^2 u}}{{\partial x^2}} \)

Laplace’s Equation:

\( \nabla^2 u = \frac{{\partial^2 u}}{{\partial x^2}} + \frac{{\partial^2 u}}{{\partial y^2}} = 0 \)