Most useful calculus formulas from BASIC to ADVANCED
Basic Calculus Formulas:
Limits :
\(\lim_{x \to a} f(x) = L\), where \(f(x)\) approaches \(L\) as \(x\) approaches \(a\).
\(\lim_{x \to a} \frac{f(x) – f(a)}{x – a} = f'(a)\) (Definition of the derivative).
\(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\).
\(\lim_{x \to 0} \frac{1 – \cos(x)}{x} = 0\).
Derivative Formulas :
Power Rule: \(\frac{d}{dx}[x^n] = n x^{n-1}\).
Constant Rule: \(\frac{d}{dx}[c] = 0\), where \(c\) is a constant.
Constant Multiple Rule: \(\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)\).
Sum/Difference Rule: \(\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\).
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: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).
Integration Formulas :
Power Rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).
Constant Multiple Rule: \(\int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx\).
Sum/Difference Rule: \(\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx\).
Integration of \(e^x\): \(\int e^x \, dx = e^x + C\).
Integration of \(\frac{1}{x}\): \(\int \frac{1}{x} \, dx = \ln|x| + C\).
Intermediate Calculus Formulas:
Trigonometric Derivatives :
\(\frac{d}{dx}[\sin(x)] = \cos(x)\).
\(\frac{d}{dx}[\cos(x)] = -\sin(x)\).
\(\frac{d}{dx}[\tan(x)] = \sec^2(x)\).
\(\frac{d}{dx}[\sec(x)] = \sec(x) \tan(x)\).
\(\frac{d}{dx}[\csc(x)] = -\csc(x) \cot(x)\).
\(\frac{d}{dx}[\cot(x)] = -\csc^2(x)\).
Trigonometric Integrals :
\(\int \sin(x) \, dx = -\cos(x) + C\).
\(\int \cos(x) \, dx = \sin(x) + C\).
\(\int \sec^2(x) \, dx = \tan(x) + C\).
\(\int \csc^2(x) \, dx = -\cot(x) + C\).
\(\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C\).
\(\int \csc(x) \, dx = \ln|\csc(x) – \cot(x)| + C\).
Fundamental Theorem of Calculus :
If \(f(x)\) is continuous on \([a, b]\), then \(\int_a^b f(x) \, dx = F(b) – F(a)\), where \(F(x)\) is an antiderivative of \(f(x)\).
Partial Derivatives :
\(\frac{\partial}{\partial x}[f(x, y)] = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) – f(x, y)}{\Delta x}\).
\(\frac{\partial}{\partial y}[f(x, y)] = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) – f(x, y)}{\Delta y}\).
Multiple Integrals :
Double Integral: \(\int_a^b \int_c^d f(x, y) \, dy \, dx\).
Triple Integral: \(\int_a^b \int_c^d \int_e^f f(x, y, z) \, dz \, dy \, dx\).
Advanced Calculus Formulas:
Taylor and Maclaurin Series :
Taylor Series: \(f(x) = f(a) + f'(a)(x-a) + \frac{f”(a)}{2!}(x-a)^2 + \cdots\).
Maclaurin Series: A special case of Taylor series around \(a = 0\), \(f(x) = f(0) + f'(0)x + \frac{f”(0)}{2!}x^2 + \cdots\)
Example: \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\).
L’Hopital’s Rule :
If \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\) (provided the limit on the right side exists).
Implicit Differentiation :
If \(F(x, y) = 0\), then \(\frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}\).
Vector Calculus :
Gradient: \(\nabla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\).
Divergence: \(\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}\) for a vector field \(\mathbf{F} = (F_1, F_2, F_3)\).
Curl: \(\nabla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} – \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} – \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} – \frac{\partial F_1}{\partial y} \right)\).
Green’s Theorem :
\(\oint_C \left( P(x, y) \, dx + Q(x, y) \, dy \right) = \iint_R \left( \frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y} \right) dA\), where \(C\) is a positively oriented, piecewise smooth curve, and \(R\) is the region bounded by \(C\).
Stokes’ Theorem :
\(\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\), where \(C\) is a closed curve, and \(S\) is the surface bounded by \(C\).
Divergence Theorem :
\(\oint_V (\nabla \cdot \mathbf{F}) \, dV = \oint_S \mathbf{F} \cdot d\mathbf{S}\), where \(V\) is a volume and \(S\) is the boundary surface of \(V\).
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