Derivatives

The derivative of a function is the slope of that function anywhere the function is well behaved. Well behaved functions are those where there is a unique value and unique slope at every point. A constant function such as y = −1 or y = 4 is a straight line of slope zero.  A linear function such as y = 2x −3 has a constant slope of 2.
The simplest function that does not have a constant slope is the quadratic function, y = x2. Considering only positive values of x, the slope increases with x. The accompanying figure shows an exploded view of a portion of the y = x2 curve with the approximate slope shown as Δy  over Δx. Starting with some arbitrary point on the curve the coordinates are x and y = x2 . Moving a small distance Δx in the x-direction the other point on the curve is x + Δx and y = (x + Δx)2. The difference in y is shown on the figure. The general expression for the slope of the curve is Δy over Δx. In mathematical form this reads

The general expression for the slope of the curve at any point x is the limit of this approximation as Δx goes to zero. Using the mathematical symbolism of limits, the general expression for the slope of y = x2 is

This defining equation is called dy/dx, with the d notation indicating the limit of Δy/Δx, or y', or f'(x).

 For this quadratic we have the following.

This general expression for the derivative is used to determine the slope of the curve y = x2 at any point.  When x = 3, the function has value 9 and slope 6. When x = 4, the function has value 16 and slope 8.

Another more general way of writing the definition is

where the expression y(x + Δx) means the value of y at x + Δx and y(x) means the value of y at x.

Polynomials  There is a pattern to the derivatives of polynomials as illustrated in the accompanying table. The pattern suggests a general rule for differentiating polynomials.

If f(x) = cxn, then  f'(x) = cnxn−1

The power law works for positive, negative and fractional exponents.

The derivative of a polynomial is accomplished by differentiating the polynomial term by term.

The derivative of y = 3x2 + x + 2 is y' = 6x + 1.

The derivative can be evaluated at any point by substituting that point value into the dy/dy equation.

The derivative can be thought of as a rate.  A most convenient way to illustrate this is with velocity and acceleration.  One of the easiest rates to visualize is velocity, distance divided by time.  If something moves 200 meters (m) in 50 seconds (s) we say it has a velocity of 4 m/s.  This 200 m in 50 s  produces an average velocity, Δx/Δt in calculus language.  The velocity at any instant during the 50 s may, however, be quite different from the average.  To find the instantaneous velocity we  first need to know how x varies with time, or x = f(t).  Then dx/dt, the limit as the time interval becomes shorter and shorter, is an expression for the instantaneous velocity, v, that can be evaluated at any time.  

If something is changing velocity as it moves then we can take the difference in velocity between the beginning and end of a time interval and calculate the average acceleration over that time interval.  The instantaneous acceleration (a = dv/dt), the rate at which the velocity changes, is the derivative of the velocity-time relation evaluated at any time.

Velocity and acceleration problems are excellent test problems.  Be sure you know that given position as a function of time x = f(t), the velocity is the first derivative, and the acceleration is the second derivative.

Given x = 4 + 6t − 5t2  know how to find velocity (v = 6 − 10t) and acceleration (a = − 10) and be able to evaluate velocity and acceleration at any time.

 The velocity is the first derivative of position, v = dx/dt.  The acceleration is the first derivative of the velocity, a = dv/dt.  Both derivatives come from the same function.  The velocity is the first derivative and the acceleration the second derivative.  It is common to write v as a first derivative and a as a second derivative:

You will encounter second and third derivatives of the same function in other areas besides velocity and acceleration. Velocity and acceleration are used as examples of this second derivative idea because they are easy to visualize.

You may want to work some problems differentiating polynomials before moving on to differentiating products and fractions.

Products and Quotients  In the case of products, differentiation is relatively simple. It is the first times the derivative of the second and the second times the derivative of the first. In mathematical symbolism product differentiation is:

If  f(x) = u(x)v(x)  then  f'(x) = u(x)v'(x) + u'(x)v(x)

For fractions differentiation is somewhat more difficult. The general rule is:

Trigonometric Functions  There is no general rule for determining the derivative of trigonometric functions.  Each trigonometric functions has a unique derivative. A table of mathematical functions or the table in the back of your text will have the derivative of common trigonometric functions.

Implicit Differentiation   The general procedure for differentiating a polynomial function y = x2 + 2x is to apply the power law rule to each term and write dy/dx = 2x + 2.  Another and often very convenient way of looking at the problem would be to differentiate the entire equation term by term,  dy = 2xdx + 2dx, and then write dy/dx = 2x + 2.  You should notice that most differential tables are written in this manner.  As functions become more complicated implicit differentiation becomes more convenient. Suppose you have a function x3 + x2y2 − xy4 = 12 where it is impossible to solve for x in terms of y or y in terms of x.  Implicit differentiation is the only way to find dy/dx.

Change of Variable  A change of variable is very convenient in differentiating certain functions. For instance if you want to differentiate y = (x2 + 2)21 you could expand the function and differentiate term by term or you view (x2 + 2) as a variable and apply the power law to get dy = 21(x2+ 2)21 d(x2 + 2)  and dy/dx = 42x(x2+ 2)21. The change in variable technique will be very valuable in the next section dealing with chain derivatives.

Chain Rule  In many practical situations a quantity is given in terms of a variable and then this variable is expressed in terms of a third variable.  A problem may be described this way because the first variable is not easily written in terms of the third or perhaps it is conceptually easier to understand the process in two steps.

Suppose the cost of manufacturing a certain item, say a computer chip, depends on the number of items produced.  The number of items produced depends on the length of time the "fab" operates to produce the chips, the length of time for the production run.  If the cost per unit (dollars per chip) is dC/dN and the rate of production (chips per hour) is dN/dt, then cost per unit of time is the product of these two derivatives.

Logarithms and Exponents  Logarithms and exponents have unique derivatives. The laws governing the derivatives of logs and exponents are listed below:

If  y = ex  then  dy = exdx

If  y = ax  and  a > 0  and ≠ 1  then  dy = (lna)axdx

If  y = ln x  then  dy = (1/x)dx

If  y = loga x   and   a > 0  and   ≠ 1  then  dy = (1/lna) x dx

The following problems will help you to fully understand and be able to apply differentiation.

TDE01 Derivative of a cubic

DE02 Derivative of a polynomial

TDE03 Derivative of a polynomial

TDE04 Evaluate a derivative

DE10 Manufacturing cost

TDE11 Instantaneous velocity

TDE12 Instantaneous acceleration

DE21 Product rule

DE25 Quotient rule

DE30 Trigonometric differentials

DE35 Implicit differential

DE40 Change of variable

DE45 Chain rule

DE50 Exponential derivative

DE55 Logarithmic derivative