Verified formula topic
Calculus Formulas
Browse 5 reviewed calculus formulas with concrete inputs, outputs, calculations, applicability checks, and LaTeX.
Calculus
Chain Rule
\frac{d}{dx}f(g(x))=f'(g(x))g'(x)Differentiates a composite function by combining outer and inner derivatives.
Open formulaCalculus
Derivative Definition
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}Defines the instantaneous rate of change of a function as a limit.
Open formulaCalculus
Fundamental Theorem of Calculus
\int_a^b f(x)\,dx=F(b)-F(a),\quad F'=fConnects definite integration with antiderivatives.
Open formulaCalculus
Integration by Parts
\int u\,dv=uv-\int v\,duTransforms an integral of a product using the product rule in reverse.
Open formulaCalculus
Product Rule
(fg)'=f'g+fg'Differentiates the product of two differentiable functions.
Open formulaBrowse 16 supporting references
Calculus
Curl
\nabla\times\mathbf FMeasures local rotation of a three-dimensional vector field.
Open formulaCalculus
Derivative Chain Rule
\frac{d}{dx}f(g(x))=f\prime(g(x))g\prime(x)Differentiates a composite function.
Open formulaCalculus
Derivative Power Rule
\frac{d}{dx}x^n=nx^{n-1}Differentiates a power of x.
Open formulaCalculus
Derivative Power Rule
\frac{d}{dx}x^n=nx^{n-1}Differentiates a power of x.
Open formulaCalculus
Derivative Product Rule
(fg)\prime=f\prime g+fg\primeDifferentiates the product of two differentiable functions.
Open formulaCalculus
Derivative Quotient Rule
\left(\frac fg\right)\prime=\frac{f\prime g-fg\prime}{g^2}Differentiates the quotient of two functions.
Open formulaCalculus
Derivative of Cosine
\frac{d}{dx}\cos x=-\sin xGives the derivative of the cosine function.
Open formulaCalculus
Derivative of Exponential
\frac{d}{dx}e^x=e^xShows that the natural exponential is its own derivative.
Open formulaCalculus
Derivative of Natural Logarithm
\frac{d}{dx}\ln x=\frac{1}{x}Differentiates the natural logarithm on its real domain.
Open formulaCalculus
Derivative of Sine
\frac{d}{dx}\sin x=\cos xGives the derivative of the sine function.
Open formulaCalculus
Divergence
\nabla\cdot\mathbf F=\sum_i\frac{\partial F_i}{\partial x_i}Measures local source or sink strength of a vector field.
Open formulaCalculus
Gradient
\nabla f=\left(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n}\right)Forms a vector of partial derivatives of a scalar field.
Open formulaCalculus
Integral Power Rule
\int x^n\,dx=\frac{x^{n+1}}{n+1}+CIntegrates a power of x when n is not −1.
Open formulaCalculus
Integral of Exponential
\int e^x\,dx=e^x+CIntegrates the natural exponential function.
Open formulaCalculus
Quotient Rule
\left(\frac{f}{g}\right)\prime=\frac{f\prime g-fg\prime}{g^2}Differentiates a quotient of two functions.
Open formulaCalculus
Taylor Series
f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^nRepresents a sufficiently smooth function as a power series around a point.
Open formula