Primer on Common Derivatives and Integrals

Table of Contents

1. Basic Derivatives

Function f(x)f(x)Derivative f(x)f'(x)
Constant cc00
xnx^nnxn1n x^{n-1}
1x\frac{1}{x}1x2-\frac{1}{x^2}
ecxe^{cx}cecxce^{cx}
axa^xaxln(a)a^x \ln(a)
ln(x)\ln(x)1x\frac{1}{x}
loga(x)\log_a(x)1xln(a)\frac{1}{x \ln(a)}
sin(x)\sin(x)cos(x)\cos(x)
cos(x)\cos(x)sin(x)-\sin(x)
tan(x)\tan(x)sec2(x)\sec^2(x)

2. Common Derivative Rules


3. Common Special Derivatives


4. Derivatives as Limits

f(x)=limh0f(x+h)f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

5. Basic Integrals

Function f(x)f(x)Indefinite Integral f(x)dx\int f(x) \, dx
00CC (arbitrary constant)
xnx^nxn+1n+1+C\frac{x^{n+1}}{n+1} + C (for n1n \neq -1)
1x\frac{1}{x}ln(x)+C\ln( \mid x \mid) + C
ecxe^{cx}1cecx+C\frac{1}{c}e^{cx} + C
axa^xaxln(a)+C\frac{a^x}{\ln(a)} + C
sin(x)\sin(x)cos(x)+C-\cos(x) + C
cos(x)\cos(x)sin(x)+C\sin(x) + C
sec2(x)\sec^2(x)tan(x)+C\tan(x) + C
csc2(x)\csc^2(x)cot(x)+C-\cot(x) + C
sec(x)tan(x)\sec(x)\tan(x)sec(x)+C\sec(x) + C
csc(x)cot(x)\csc(x)\cot(x)csc(x)+C-\csc(x) + C

6. Common Integral Rules


7. Integration by Parts

Example: Let u=xu = x and dv=exdxdv = e^x \, dx. Then, du=dxdu = dx and v=exv = e^x. Thus,

xexdx=xexexdx=xexex+C\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C

where CC is the constant of integration.


8. Gradient, Divergence, Curl and Laplacian

Gradient

The gradient of a scalar function f(x,y,z)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:

f=(fx,fy,fz)\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.

The function

f(x,y)=x2+y2f(x, y) = x^2 + y^2

has gradient

f=(2x,2y)\nabla f = (2x, 2y)

which can be plotted as a vector field.

Divergence

The divergence of a vector field F=(Fx,Fy,Fz)\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:

F=Fxx+Fyy+Fzz\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.

The vector field

F(x,y)=(x2,y2)\mathbf{F}(x, y) = (x^2, -y^2)

has divergence

F=2x2y\nabla \cdot \mathbf{F} = 2x - 2y

which can be plotted as a scalar field.

Curl

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

×F=(FzyFyz,FxzFzx,FyxFxy)\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.

The vector field

F(x,y)=(y2,x2)\mathbf{F}(x, y) = (-y^2, x^2)

has curl

×F=(0,0,2x+2y)\nabla \times \mathbf{F} = (0, 0, 2x + 2y)

which can be plotted as a vector field.

Laplacian

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

2f=2fx2+2fy2+2fz2\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.

The function

f(x,y)=sin(x)+cos(y)f(x, y) = \sin(x) + \cos(y)

has Laplacian

2f=sin(x)cos(y)\nabla^2 f = -\sin(x) - \cos(y)

which can be plotted as a scalar field.