Common derivative formulas

The derivative of a function $f(x)$ is defined as:

$$\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

First derivatives of basic functions

Constant:

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

Power rule:

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

Linear function:

$$\frac{d}{dx}(a x + b) = a$$

Exponential:

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

Natural logarithm:

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

Trigonometric derivatives

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

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

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

Higher-order derivatives

Second derivative:

$$\frac{d^2 f}{dx^2} = \frac{d}{dx}\left(\frac{df}{dx}\right)$$

Example:

$$f(x) = x^3 \quad\Rightarrow\quad \frac{d^2 f}{dx^2} = 6x$$

Product rule

For $f(x)g(x)$:

$$\frac{d}{dx}\left(fg\right) = f' g + f g'$$

Quotient rule

For $\frac{f(x)}{g(x)}$:

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

Chain rule

For $f(g(x))$:

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

Partial derivatives

For a multivariable function $f(x,y)$:

$$\frac{\partial f}{\partial x}, \qquad \frac{\partial f}{\partial y}$$

Example:

$$f(x,y) = x^2 y$$

$$\frac{\partial f}{\partial x} = 2xy, \qquad \frac{\partial f}{\partial y} = x^2$$

Derivatives commonly used in beam theory

Deflection:

$$w'(x) = \frac{dw}{dx}$$

Rotation:

$$\theta(x) = w'(x)$$

Curvature:

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

Bending moment relation:

$$M(x) = -E I\,\frac{d^2 w}{dx^2}$$

Shear force relation:

$$V(x) = \frac{dM}{dx}$$

Distributed load relation:

$$q(x) = -\frac{dV}{dx}$$