Here are some basic laws which can be used to derive other differentiation rules. 
The derivative of a constant is zero.

If a function does not vary (is constant), its rate of change is zero.
More formally, if y = c (a constant), then 

dy/dx = d/dx (c) = 0

Example: 
y = 5
dy/dx = d/dx (5) = 0

The derivative of the product of a constant and a function is equal to the constant times the derivative of the function.

That is, if y = c u, then 

dy/dx = d/dx (c × u)=c x d/dx (u)

y = 8 × x
dy/dx = d/dx (8 × x)
dy/dx= 8 × d/dx (x)
dy/dx= 8 × (1) = 8

The derivative of the sum or difference of two functions is equal to the sum or difference of the derivatives of the functions

y = u + v

dy/dx =d/dx(u) + d/dx(v)

Example:
y = 8 × x - x^2
dy/dx = d/dx (8 × x - x^2)
dy/dx= 8 × d/dx (x) - d/dx (x^2)
dy/dx= 8 - 2 × x 

The derivative of the product of two functions is equal to the first function times the derivative of the second, plus the second function times the derivative of the first

That is, if y = u v, then 

dy/dx =d/dx(uv) = v du/dx + u dv/dx 

Example:
y = x ex
dy/dx = d/dx (x ex) = x d/dx (ex) + ex d/dx (x)
dy/dx = x ex + ex = ex (x + 1) 

The derivative of the quotient of two functions, is equal to the function in the denominator times the derivative of the function in the numerator, minus the function in the numerator times the derivative of the function in the denominator, all divided by the square of the function in the denominator

That is, if y = u/v, then 

dy/dx =d/dx(u/v) = (v du/dx - u dv/dx)/ v^2

We can find the derivative of the quotient of two functions in a different way. If we consider the equation y = u/v as y = u v-1 and then apply the product rule of derivatives (above) we get the same result. 
Example:
y = ex/x 
y = ex × x-1
Now applying the product rule, we have,
dy/dx = ex d/dx (x-1) + x-1 d/dx (ex)
dy/dx = ex (-1)x-2 + x-1 ex
dy/dx = ex(- x-2 + x-1)