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Chain Rule

\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)

Derivative of a composite function.

Product Rule

(fg)' = f'g + fg'

Derivative of a product of two functions.

Quotient Rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Derivative of a quotient of two functions.

Derivative of Sine

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

Derivative of the sine function.

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

Derivative of the cosine function.

Derivative of e^x

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

The exponential function is its own derivative.

Derivative of ln x

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

Derivative of the natural logarithm.

Power Rule (Integral)

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \ne -1

Antiderivative of a power function.

CalculusAdvanced

Derivative (Limit Definition)

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

Formal definition of the derivative.

Constant Rule (Derivative)

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

The derivative of any constant is zero.

Sum Rule (Derivative)

(f+g)' = f' + g'

The derivative of a sum is the sum of the derivatives.

Constant Multiple Rule

\frac{d}{dx}(c\,f) = c\,f'

Constants factor out of a derivative.

Second Derivative

f''(x) = \frac{d}{dx}\left(f'(x)\right)

The derivative of the first derivative; measures concavity.

Derivative of Tangent

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

Derivative of the tangent function.

Power Rule

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

Derivative of a power function.

Product Rule

(uv)' = u'v + uv'

Derivative of a product of two functions.

Quotient Rule

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Derivative of a quotient of two functions.

Chain Rule

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

Derivative of a composite function.

Derivative of Sine

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

The derivative of sine is cosine.

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

The derivative of cosine is negative sine.

Derivative of Exponential

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

The exponential function is its own derivative.

Derivative of Natural Log

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

The derivative of the natural logarithm.

Power Rule for Integration

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Antiderivative of a power function (n≠-1).

Integral of Exponential

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

Antiderivative of the exponential function.

Integral of 1/x

\int \frac{1}{x}\,dx = \ln|x| + C

Antiderivative of the reciprocal function.

Data Science & MLIntermediate

Second Derivative Test

f''(x) > 0 \Rightarrow \text{min}

Concavity determines local minima and maxima.

CalculusAdvanced

Partial Derivative

\frac{\partial f}{\partial x}

Rate of change with respect to one variable, holding others fixed.

CalculusAdvanced

Gradient Vector

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

Vector of partial derivatives pointing in the steepest ascent.

Sigmoid Function

\sigma(x) = \frac{1}{1 + e^{-x}}

Maps any real number to the range (0, 1); used in logistic regression.

Data Science & MLAdvanced

Softmax Function

\sigma(z_i) = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}

Converts a vector of scores into a probability distribution.

Data Science & MLBasic

ReLU Activation

f(x) = \max(0, x)

Rectified Linear Unit: outputs the input if positive, else zero.

Physics Laws & TheoremsBasic

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum of an isolated system stays constant.

Conservation of Energy

E_{total} = \text{const}

Energy cannot be created or destroyed, only transformed.

Physics Laws & TheoremsAdvanced

Bernoulli's Principle

P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Faster-moving fluid has lower pressure (energy conservation in flow).

First Law of Thermodynamics

\Delta U = Q - W

Energy conservation: change in internal energy equals heat added minus work done.

Physics Laws & TheoremsBasic

Kirchhoff's Current Law

\sum I_{in} = \sum I_{out}

Current into a junction equals current out of it (charge conservation).

Conservation of Energy

E_{total} = \text{constant}

Energy cannot be created or destroyed, only transformed.

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum is conserved without external forces.