Derivative Summary

Summary of the Basic Derivative Rules

Here we summarize all of the basic rules you should know for finding derivatives of functions. [Actually, we will not be using trig functions in this course; but the calculus rules for them will be given for completeness.] If the goal were only to be able to efficiently calculate the derivative of a function then you should memorize all of these rules. However, in order to fully understand derivatives you should also know how these rules are related and how to derive some of the rules from others. These derivations give you techniques and facts that can be useful in other contexts. This section shows how to use the rules given in bold to derive the other rules.

Note that you should not forget about the definition and interpretation of the meaning of a derivative. It is an instantaneous rate of change. So the derivative functions that are given below can be used to find this rate of change at every value of x in the domain of the function. Also, the formal definition of a derivative can be described as the limit of an average rate of change between two points as the difference between them goes to zero. Formal proofs of the rules given below must verify the formal definition. Such proofs will not be discussed here; but can be found in any standard calculus textbook.

Derivatives of Basic Functions
	Function	Its Derivative (with respect to x)
1.	cxⁿ	cnx^n-1
2.	sin(x)	cos(x)
3.	cos(x)	-sin(x)
4.	e^x	e^x
5.	a^x	[ln(a)]a^x	how to derive
6.	ln(x)	1/x
7.	log_a(x)	1 / ([ln(a)]x)	how to derive

Notes:

Rule 1 includes several rules that most textbooks list separately. If n = 0 then the function is a constant and its derivative is zero. When n = 1 the function is linear and you should know that the derivative of a linear function is its slope, which is c in this case.
Other trig functions are not listed since they can be derived from sine and cosine using the rules in the "Combinations of Functions" table below. Also, sine and cosine are much more common than the other functions. Here is how to derive the other trig functions.
Rules 4 and 6 are special cases of Rules 5 and 7, respectively, when a = e. So one could memorize just Rules 5 and 7. However, Rules 4 and 6 are simpler and the "change of base" technique used in the derivation of Rules 5 and 7 is useful to know. Also, base e exponentials and natural logarithms are more commonly used in applications. [Although there are some important computer science uses of log₂].
Actually, if the goal is to efficiently calculate derivatives then you are probably better off memorizing the "chain rule" versions of these rules. This table gives the chain rule versions of all the derivative rules for basic functions. These are the versions you will probably want to memorize. The "chain rule" versions of the rules replace the variable x with u, which is a function of x. So all of the derivatives have u in place of x and are multiplied by the derivative of u. This is just including an implicit use of the chain rule (Rule 6' given below). I have chosen not to present this material in that way so that it will be more likely that you will realize when you are using (or need to use) the chain rule, which is a very important rule in calculus.

We also have rules for finding the derivatives of combinations of the above basic functions. For example we can add (subtract) or multiply (divide) functions together; take the composition of two functions, and take the inverse of a function.

Derivatives of Combinations of Functions
	Combination of Functions	Its Derivative (with respect to x)
1'.	f + g	f ′ + g′
2'.	cf	cf ′	how to derive
3'.	f g	f ′ g + g′ f
4'.	f/g	[f ′ g - g′ f ]/g²	how to derive
5'.	fⁿ	n f^n-1 f ′	how to derive
6'.	g(f)	g′ (f) f ′
7'.	inverse of f, i.e. f^-1	1 / f ′(f^-1(x)) where x is in the domain of f^-1	how to derive

Note: f and g represent basic functions; c and n represent real constants.

Finally, the rules in the "Combination of Functions" table are given in words and other symbols. The following uses D_x f to mean df /dx.

Derivatives of Combinations of Functions
1'. Sum Rule	Derivative of the sum of functions is the sum of the derivatives, i.e. D_x[f + g] = D_xf + D_xg.
2'. Scalar Multiplication Rule	Derivative of a constant times a function is the constant times the derivative of the function, i.e. D_x[cf] = cD_xf.
3'. Product Rule	Derivative of the product of two functions is the derivative of the first times the second plus the derivative of the second times the first, i.e. D_x[fg] = g D_xf + f D_xg.
4'. Quotient Rule	Derivative of the quotient of two functions is the derivative of the numerator times the denominator minus the derivative of the denominator times numerator all divided by the square of the denominator, i.e. D_x[f/g] = [g D_xf - f D_xg]/[g²]
5'. Power Rule	Derivative of a function raised to the nth power is the derivative of the function times n times the function raised to the (n-1)st power, i.e. D_x[f ⁿ] = n f ^n-1 D_xf
6'. Chain Rule	Derivative of the composition of functions is the product of the derivatives of the functions (evaluated at the appropriate points, i.e. D_x[f o g] = D_tf(g(x)) D_xg(x), where t = g(x) and y = f(t). This is the Chain Rule, which can also be written in this short form : dy/dx = [dy/dt] [dt/dx]
7'. Inverse Function Rule	Derivative of the inverse of a function is the reciprocal of the derivative of the function, where both derivatives are evaluated at the appropriate values, i.e. if f and g are inverses the f ′(x) = 1/g′(f(x)) where x is in the domain of f. This follows from applying the chain rule to differentiate both sides g(f(x)) = x, which is a property of inverses.