Notation: X,Y=variables; f,g,u,v=functions of X; b,n=numerical parameters
 

Rule (click on Rule number to see proof)	Function	Derivative
Rule 1: Rule of constants	Y=b	dY/dX = 0
Rule 2: Power rule	Y = Xn	dY/dX = nXn-1
Rule 3: Rule of coefficients	Y = b u(X)	dY/dX = b du/dX
Rule 4: Summation rule	Y = u + v	dY/dX = du/dX + dv/dX
Rule 5: Product rule	Y = u v	dY/dX = u dv/dX + v du/dX
Rule 6: Quotient rule	Y = u/v	v du/dX - u dv/dX  dY/dX =  ------------------                              v2
Rule 7: Chain rule	Y = f(u(X))	dy/dX = dY/du du/dX

 
The rules in words:
    Rule 1: Rule of constants: the derivative of a constant is zero.
    Rule 2: Power rule: the derivative of a power function equals the value of the exponent times X raised to the exponent minus one.
    Rule 3: Coefficients rule: the derivative of a coefficient times a function equals the coefficient times the derivative of the function.
    Rule 4: Summuation rule: the derivative of the sum of a finite number of functions equals the sum of their derivatives.
    Rule 5: Product rule: the derivative of the product of two functions equals the first function times the derivative of the second function plus the second function times the derivative of the first function.
    Rule 6: Quotient rule: the derivative of the quotient of two functions equals the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
    Rule 7: Chain rule: suppose Y is a function of another function (called a nested function) which in turn is a function of X. Then the derivative of Y with respect to X is the derivative of Y with respect to the nested function times the derivative of the nested function with respect to X.

Addendum: Implicit functions
    An explicit function of X has the form: Y = f(X) You can apply the 7 rules directly to explicit functions.
    Any other arrangement of terms is called an implicit function of X. These usually have the form: f(X,Y) = b, where b is a constant. You can also apply the 7 rules above to implicit functions, as follows:
    1. Treat both sides of the equation as explicit functions and take the derivatives of both sides. Since the righthand side is a constant, its derivative will equal zero.
    2. Anytime you run into an expression containing Y on the lefthand side, say g(Y), just apply the chain rule: dg/dx = dg/dy dy/dx.
    3. After differentiating, rearrange terms to isolate dy/dx.