Primer on Common Derivatives and Integrals

Table of Contents

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1. Basic Derivatives

Function $f(x)$	Derivative $f'(x)$
Constant $c$	$0$
$x^n$	$n x^{n-1}$
$\frac{1}{x}$	$-\frac{1}{x^2}$
$e^{cx}$	$ce^{cx}$
$a^x$	$a^x \ln(a)$
$\ln(x)$	$\frac{1}{x}$
$\log_a(x)$	$\frac{1}{x \ln(a)}$
$\sin(x)$	$\cos(x)$
$\cos(x)$	$-\sin(x)$
$\tan(x)$	$\sec^2(x)$

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2. Common Derivative Rules

• Sum Rule

  $$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$

• Product Rule

  $$\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

• 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

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

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3. Common Special Derivatives

• $$\frac{d}{dx} (\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$$
• $$\frac{d}{dx} (\cos^{-1}(x)) = \frac{-1}{\sqrt{1-x^2}}$$
• $$\frac{d}{dx} (\tan^{-1}(x)) = \frac{1}{1+x^2}$$

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4. Derivatives as Limits

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

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5. Basic Integrals

Function $f(x)$	Indefinite Integral $\int f(x) \, dx$
$0$	$C$ (arbitrary constant)
$x^n$	$\frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
$\frac{1}{x}$	$\ln( \mid x \mid) + C$
$e^{cx}$	$\frac{1}{c}e^{cx} + C$
$a^x$	$\frac{a^x}{\ln(a)} + C$
$\sin(x)$	$-\cos(x) + C$
$\cos(x)$	$\sin(x) + C$
$\sec^2(x)$	$\tan(x) + C$
$\csc^2(x)$	$-\cot(x) + C$
$\sec(x)\tan(x)$	$\sec(x) + C$
$\csc(x)\cot(x)$	$-\csc(x) + C$

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6. Common Integral Rules

• Sum Rule:

  $$\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$$

• Constant Multiple Rule:

  $$\int c f(x) \, dx = c \int f(x) \, dx$$

• Substitution Rule (u-substitution): If $u = g(x)$, then

  $$\int f(g(x))g'(x) \, dx = \int f(u) \, du$$

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7. Integration by Parts

• Integration by Parts Formula: $$\int u \, dv = uv - \int v \, du$$ where $u$ and $dv$ are chosen parts of the integrand. This method is useful when the integrand is a product of two functions. The choice of $u$ and $dv$ can significantly affect the complexity of the resulting integral.

Example: Let $u = x$ and $dv = e^x \, dx$. Then, $du = dx$ and $v = e^x$. Thus,

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

where $C$ is the constant of integration.

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8. Gradient, Divergence, Curl and Laplacian

Gradient

The gradient of a scalar function $f(x, y, z)$ is a vector field that points in the direction of the greatest rate of increase of the function. It is denoted as:

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

where $\nabla$ is the del operator. The gradient is used in optimization problems to find local maxima and minima of functions.

Divergence

The divergence of a vector field $\mathbf{F} = (F_x, F_y, F_z)$ is a scalar function that measures the rate at which "stuff" is expanding or contracting at a point. It is given by:

$$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$

Divergence is used in fluid dynamics to describe the behavior of fluid flow.

Curl

The curl of a vector field $\mathbf{F} = (F_x, F_y, F_z)$ is a vector that describes the rotation or "twisting" of the field. It is given by:

$$\nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)$$

Curl is used in electromagnetism and fluid dynamics to describe rotational motion.

Laplacian

The Laplacian of a scalar function $f(x, y, z)$ is a scalar that measures the "curvature" or "spread" of the function. It is given by:

$$\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$$

The Laplacian is used in physics to describe diffusion and wave propagation.