Calculus Basic Rules

Definitions

• Differentiation is finding the slope of a function
• Integration is finding the area described by the differential of a function
• Logic check:
• Integrating a differential must produce the original function
• Plus a constant, possibly
• The integration produces a whole class of functions considering the unknown added constant
• Differentiating an integral must produce the original differential 

Types of Functions

Case	Function Type	Example	Notes
Linear	$$x$$	$$10^3$$
Polynomial	$$x^2, x^3, . . . x^n$$	$$10^6, 10^9, ... 10^{3n}$$
Exponential	$$2^x, e^x, 10^x$$	$$10^{300}, 10^{434}, 10^{1,000}$$
Factorial	$$x! , x^x$$	$$10^{2,566}, 10^{3,000}$$	$$x! = \dfrac{x^x}{e^x}$$

Simple Differentiation/Integration - The Power Rule

• $ \mbox{Function to differeniate is } f(x) = x^n  $
• $ \dfrac{d}{dx}\left(x^n\right) = n x^{n-1} $
• $ \mbox{Example:} $
• $f(x) =  x^5 $
• $\dfrac{d}{dx} f(x) = 5x^4  $
• $ \mbox{Function to integrate}= m x^{n-1} $
• $\int{m x^n}\,dx = \dfrac{m x^{n+1}}{n+1} + constant $
• $ \mbox{Example:}$
• $f(x) = 5 x^4$
• $\int{5x^4}  = \dfrac{5x^{4+1}}{5} + constant =x^5 + constant $

Differention/Integration of two added functions

• $\mbox{Function to differentiate is } p(x) = a f(x) + b g(x) $
• $ \dfrac{dp}{dx} = a \dfrac{df}{dx} + b \dfrac{dg}{dx}  $ 
• $\mbox{Function to integrate} = a \dfrac{df}{dx} + b \dfrac{dg}{dx}  $ 
• $\mbox{Example:}$
• $f(x) = x^2 + x^3 $
• $\int(x^2 + x^3) \, dx = \int(x^2)\,dx + \int(x^3)\,dx = \dfrac{x^3}{3} + \dfrac{x^4}{4} $

Differentiation/Integration of two multiplied functions

• $ \mbox{Function to differeniate is } p(x) = f(x) g(x)  $
• $ \dfrac{dp}{dx} = f(x) \dfrac{dg}{dx} +  g(x) \dfrac{df}{dx}  $
• $ \mbox {Example:}$
• $f(x) = x^5 sin{(x)}$
• $\dfrac{d}{dx} f(x)= 5x^4 sin{(x)} + x^5 cos{(x)} $
• $ \mbox{Function to integrate}= f(x) \dfrac{dg}{dx} +  g(x) \dfrac{df}{dx}  $
• 
  

Differentiation/Integration of two divided functions

• $ \mbox{Function to differentiate is}\; p(x) = \dfrac{f(x)}{g(x)} $
• $ \dfrac{dp}{dx} = \dfrac{(g(x) \dfrac{df}{dx} - f(x) \dfrac{dg}{dx})}{g^2} $

Differentiation/Integration of nested functions

• $ \mbox{Function to differeniate is } p(x) = f(g(x))$
• $\mbox{so let }$
• $ y = g(x) $
• $ z = f(y) \\  $
• $ \dfrac{dz}{dx} = \dfrac{dz}{dy} *  \dfrac{dy}{dx}  $
• $ \mbox{Example:} \; f(x) = \sin(3x) $
• $ z = \sin(y) \; \mbox{and} \; y = 3x $
• $\dfrac{dz}{dx} =  \cos(y) * 3 $
• $\mbox{Now substitute for y} $
• $\dfrac{dz}{dx} =  3 cos(3x) $
• $ \mbox{Function to integrate}= 3 cos(3x) $
•  

Differentiation of Inverse Function

• $\mbox{If}\;y = f(x)\;\mbox{then}\; x = f^-1 (y) $
• $\mbox{Example:}$
• $\mbox{If the original function is}\; y = e^x$
• $\mbox{Then the inverse function is}\; x = \ln(y)$
• $\mbox{The relationship of the differentials of the original function and its inverse are:}$
• $\dfrac{dx}{dy} = \dfrac{1}{\dfrac{dy}{dx}} $

Fundamental Theorem of Calculus

• For a function f(x)
• Find a function F(x) whose derivative is f(x)
• Then while f(x) is positive
• $\int^B_A f(x) \, dx = F(x=B) - F(x=A) $

Table of Differentials & Integrals

Differential	Function	Integral	Comments
$$n x^{n-1} $$	$${x^n}$$	$$\left(\dfrac{x^{n+1}}{n+1}\right) + C$$	$$n \ne -1 $$
$$\dfrac{1}{x}$$	$$\ln{x}$$	$$(x \ln{x}) - x  $$
$$ce^{cx}$$	$${e^{cx}}$$	$$\dfrac{e^{cx}}{c}$$
$$\cos{x}$$	$${\sin{x}} $$	$$-\cos{x} $$
$$-\sin{x}$$	$${\cos{x}} $$	$$\sin{x} $$
	$${\tan{x}} $$	$$\ln{\sec{x}} $$
	$${\csc{x}} $$	$$-\ln{(\csc{x} + \tan{x})} $$
	$${\csc} $$	$$\ln\tan\dfrac{x}{2} $$
	$${} $$	$$ $$

Examples of practical uses of Differentiation

• Note:
• The first derivative when set to zero calculates a minimum or a maximum or both. 
• The second derivative tells us how the first derivative changes. 
• If it is always less than zero, then the results from the first derivative = 0 calculations is a maximum. 
• If it is always greater than zero, then the results from the first derivative = 0 calculations is a minimum.
• Example:
• $y = x^2 \; \mbox{is a parabola with a minimum at x = 0.}$
• $\dfrac{d(x^2)}{dx}  = \dot{x^2} = 2 x $
• $ 2x = 0 \therefore x = 0 \; \mbox{is a min or a max.}  $
• $ \dot{2 x} = \ddot{x^2} = 2 \; \mbox{thus x = 0 is a min, i.e. all changes to slope go up} $
• A rectangular storage area is to be constructed along the side of a tall building. A security fence is required along the remaining 3 sides of the area. What is the maximum area that can be enclosed with 800 m of fencing?
  
  Answer
  
  
  The area is A=xy
  We know 2x+y=800 so y=800−2x
  So the area is A = x(800 − 2x) = 800x − 2x²
  To maximise the area, find when dAdx=0
  

  dAdx=800−4x=0

  when
  

  x=200

  Is it a maximum?
  

  d2Adx2=−4<0 for all x

  So it is a maximum.
  So the maximum area occurs when x=200, y=400 and that area is:
  A = 200 × 400 = 80 000 m² 
• A box with a square base has no top. If 64 cm² of material is used, what is the maximum possible volume for the box?
  
  Answer
  
  The net for this box would be:
  
  The volume of the box is V = x²y
  We are told that the surface area of the box is 64 cm². The area of the base of the box is x² and the area of each side is xy, so the area of the base plus the area of the 4 sides is given by:
  

  x² + 4xy = 64 cm²

  Solving for y gives:
  

  y=64−x24x=16x−x4

  So the volume can be rewritten:
  

  V=x2y=x2(16x−x4)=16x−x34

  Now
  

  dVdx=16−3x24

  and this is zero when
  

  x=±83√≈4.62

  (Note: The negative case has no practical meaning.)
  Is it a maximum?
  

  d2Vdx2=−3x2

  and this is negative when x is positive. So it is a MAX.
  So the dimensions of the box are:
  Base 4.62 cm × 4.62 cm and sides 2.31 cm.
  The maximum possible volume is
  V = 4.62 × 4.62 × 2.31 ≈ 49.3 cm³
  Check: Area of material:
  x² + 4xy = 21.3 + 4 × 4.62 × 2.31 = 64
  Checks OK.

Examples of practical uses of Integration

• $\mbox{The circumfrence of a circle} = 2 \pi r    $
• $\mbox{So the area of a circle} = \int(2 \pi r)\,dr = \pi r^2    $
• $\mbox{The surface area of a sphere} = 4 \pi r^2 $
• $\mbox{So the volume of a sphere} = \int(4 \pi r^2)\,dr = \dfrac{4}{3}\pi r^3    $

References

• Calculus lectures by MIR Professor Gilbert Strang
• Table of Differentials from Math.com
• Table of Integrals from School of Mathematics in Manchester, UK
• Table of Integrals from Math.com
• Connexions Tutorial on Basic Differentiation