Derivadas de Funciones

Reglas fundamentales

Función	Derivada
$y = c \cdot x^n + C$	$y' = c \cdot n \cdot x^{n-1}$
$y = u(x) \pm v(x)$	$y' = u'(x) \pm v'(x)$
$y = u(x) \cdot v(x)$	$y' = u'(x) \cdot v(x) + u(x) \cdot v'(x)$
$y = \frac{u(x)}{v(x)}$	$y' = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{v^2(x)}$
$y = \sqrt{x}$	$y' = \frac{1}{2\sqrt{x}}$
$y = u(x)^{v(x)}$	$y' = u^{v}(x) \left( \frac{u'(x) \cdot v(x)}{u(x)} + v'(x) \cdot \ln u(x) \right)$

Derivada de una función de función

$$y = f[u(x)]$$ $$y' = f'(u) \cdot u'(x)$$ $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

---

Derivadas de funciones paramétricas

$$y = f(x), \quad \begin{cases} x = f(t) \\ y = g(t) \end{cases}$$ $$y' = \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\dot{y}}{\dot{x}}$$ $$y'' = \frac{d^2y}{dx^2} = \frac{\dot{x}\ddot{y} - \dot{y}\ddot{x}}{\dot{x}^3}$$

---

Derivadas de funciones Inversas

Si de la ecuación $y = f(x)$ se despeja $x$,resulta la función inversa $x = \varphi(y)$.

$$x = \varphi(y)$$

Example:

$$y = f(x) = \arccos x \quad \text{gives} \quad x = \varphi(y) = \cos y$$ $$f'(x) = \frac{1}{\varphi'(y)} \bigg|_{y = f(x)}$$ $$f'(x) = \frac{1}{-\sin y} = -\frac{1}{\sqrt{1 - x^2}}$$

Exponential Functions

function	derivative
$y = e^x$	$y' = e^x$, $y'' = \dots$
$y = e^{-x}$	$y' = -e^{-x}$
$y = e^{ax}$	$y' = a \cdot e^{ax}$
$y = x \cdot e^x$	$y' = e^x \cdot (1 + x)$
$y = \sqrt{e^x}$	$y' = \frac{\sqrt{e^x}}{2}$
$y = a^x$	$y' = a^x \ln a$
$y = a^{nx}$	$y' = n \cdot a^{nx} \ln a$
$y = a^{x^2}$	$y' = a^{x^2} \cdot 2x \ln a$

Trigonometric Functions

function	derivative
$y = \sin x$	$y' = \cos x$
$y = \cos x$	$y' = -\sin x$
$y = \tan x$	$y' = \frac{1}{\cos^2 x} = 1 + \tan^2 x$
$y = \cot x$	$y' = \frac{-1}{\sin^2 x} = -(1 + \cot^2 x)$
$y = a \cdot \sin(kx)$	$y' = a \cdot k \cdot \cos(kx)$
$y = a \cdot \cos(kx)$	$y' = -a \cdot k \cdot \sin(kx)$
$y = \sin^n x$	$y' = n \cdot \sin^{n-1} x \cdot \cos x$
$y = \cos^n x$	$y' = -n \cdot \cos^{n-1} x \cdot \sin x$
$y = \tan^n x$	$y' = n \cdot \tan^{n-1} x \cdot (1 + \tan^2 x)$
$y = \cot^n x$	$y' = -n \cdot \cot^{n-1} x \cdot (1 + \cot^2 x)$
$y = \frac{1}{\sin x}$	$y' = \frac{-\cos x}{\sin^2 x}$
$y = \frac{1}{\cos x}$	$y' = \frac{\sin x}{\cos^2 x}$

Logarithmic Functions

function	derivative
$y = \ln x$	$y' = \frac{1}{x}$
$y = \log_a x$	$y' = \frac{1}{x \cdot \ln a}$
$y = \ln(1 \pm x)$	$y' = \frac{\pm 1}{1 \pm x}$
$y = \ln(x^n)$	$y' = \frac{n}{x}$
$y = \ln(\sqrt{x^3})$	$y' = \frac{1}{2x}$

---

Hyperbolic Functions

function	derivative
$y = \sinh x$	$y' = \cosh x$
$y = \cosh x$	$y' = \sinh x$
$y = \tanh x$	$y' = \frac{1}{\cosh^2 x}$
$y = \coth x$	$y' = \frac{-1}{\sinh^2 x}$

---

Inverse Trigonometric Functions

function	derivative
$y = \arcsin x$	$y' = \frac{1}{\sqrt{1 - x^2}}$
$y = \arccos x$	$y' = -\frac{1}{\sqrt{1 - x^2}}$
$y = \arctan x$	$y' = \frac{1}{1 + x^2}$
$y = \operatorname{arccot} x$	$y' = -\frac{1}{1 + x^2}$
$y = \operatorname{arcsinh} x$	$y' = \frac{1}{\sqrt{x^2 + 1}}$
$y = \operatorname{arccosh} x$	$y' = \frac{1}{\sqrt{x^2 - 1}}$
$y = \operatorname{artanh} x$	$y' = \frac{1}{1 - x^2}$
$y = \operatorname{arcoth} x$	$y' = \frac{1}{1 - x^2}$