Differentiation

Definition and Notation

A derivative of a function, f(x), is a function f’(x) such that:

Limit definition of a derivative

The Derivative function is defined at x so long as the above limit exists at x. Thus, it is generally not defined at points where functions come to sharp “corners” or cusps, or if the limit is different from the right than from the left. For example, the derivative for the function below is undefined at x=2:

This discontinuous function has an undefined derivative at x=2.

This discontinuous function has an undefined derivative at x=2.

What is a derivative? The value of f’(x) is the slope of the line tangent to f(x) at x. This is the same as saying that it is the rate of change of f(x) with respect to x at x.

Besides the f’(x) notation, several equivalent notations are found in literature and textbooks to denote a derivative:

Various notations for a derivative of f(x).  All are equivalent.

Differentiation Rules

Learning to differentiate is a process of memorizing differentiation rules for various types of functions. I will list several of the most useful rules here without much explanation. If you are rusty on differentiating, I would encourage you to find a calculus text book and work odd-numbered problems until you feel comfortable.

Basic Differentiation Rules

Derivative of a Constant

Let b be a constant, then

if f(x) = b then f’(x) = 0

Derivative of a Linear Function

if f(x) = mx + b then f’(x) = m

The Power Rule

Let n be an integer, then

if f(x) = xn then f’(x) = nxn-1

Derivative of a Function Multiplied by a Constant

(d/dx)(c*f(x)) = c*(d/dx)*f(x)

Derivative of a Sum of Two Functions

The derivative of the sum of two function is the sum of the derivatives of the functions.

The Product Rule

(d/dx)(f(x)g(x))=f(x)(d/dx)(g(x)) + g(x)(d/dx)(f(x))

The Quotient Rule

Differentiating the quotient of two functions

The Reciprocal Rule

Differentiating the reciprocal of a function

The Chain Rule

(d/dx)(f(g(x))=f'(g(x))*g'(x)

Log and Exponent Functions

Base e

(d/dx)(e^x) = e^x

(d/dx)(ln x) = 1/x

Any Other Base

(d/dx)(b^x) = (ln x)(b^x)

Derivative of the log base b of x

Trigonometric Functions

You should be aware that differentiation rules exist for all six standard trig functions (sin, cos, tan, sec, csc, cot) and their inverses (sin-1, cos-1, tan-1, sec-1, csc-1, cot-1). In practice, however, you can get away with knowing the differentiation rules for sin and cos and looking up the others if you need them. Actually, if you remember your identities from high school trig, you should be able to rearrange any trigonometric function in terms of sines and cosines.

(d/dx)(sin x) = cos x

(d/dx)(cos x) = -sin x

The hardest part is remembering which one has the negative sign.

Hyperbolic Functions

I actually doubt that there will be any questions on the exam about hyperbolic functions. They have important applications (for example in infinite sequences and series, which are important in numeric analysis and algorithm design) but they are probably a little esoteric for a test that doesn’t even cover integral calculus. Still, for the sake of completeness, let me remind you of the two basic rules:

(d/dx)(sinh x) = cosh x

(d/dx)(cosh x) = sinh x

Notice that they are just like the rules for sine and cosine, but with no negative sign.


Sources

Shenk, A.  (1988).  Calculus and Analytic Geometry [4th Ed.].  Glenview, IL:  Scott Foresman and Company.

Swokowski, E., Olinick, M., Pence, D. & Cole, J.   (1994). Calculus of a Single Variable [6th Ed.].  Boston, MA: PWS.


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