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

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

Example:

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

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

Example:

y = x e^x
dy/dx = d/dx (x e^x) = x d/dx (e^x) + e^x d/dx (x)
dy/dx = x e^x + e^x = e^x (x + 1)

The derivative of a fraction, that is, 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

Actually, we can find the derivative of the quotient of two functions in an easier 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 = e^x/x
y = e^x × x^-1

Now applying the product rule, we have,

dy/dx = e^x d/dx (x^-1) + x^-1 d/dx (e^x)
dy/dx = e^x (-1)x^-2 + x^-1 e^x
dy/dx = e^x(- x^-2 + x^-1)

If the derivative of y = f(x) is dy/dx, then the derivative of the inverse function which expresses x in terms of y is given by the formula

If y = f(u) and u = g(x), that is, if y is a function of a function, then

If y = f(t) and x = g(t), that is, if y and x are related parametrically, then

Example:

y = t³ and x = 2 t²

dy/dx = (dy/dt) / (dx/dt)

dy/dt = d/dt (t³) and dx/dt = d/dt (2 t²)

dy/dt = 3 t² and dx/dt = 4 t

dy/dx = (3 t²) / 4 t

dy/dx = 3 t / 4