THE DERIVATIVE OF A CONSTANT, A VARIABLE, AND A SUM - Differentiation of Algebraic Functions - The Calculus Primer

The Calculus Primer (2011)

Part III. Differentiation of Algebraic Functions


3—1. The Derivative of a Constant. We shall now derive standard formulas for differentiating algebraic functions, using the General Rule developed in Chapter Two. With these formulas for finding the derivative of functions and various combinations of functions, it will then be possible to differentiate specific functions in later work when applying the calculus to practical problems.

The simplest algebraic function is that given by y = c, where c is any constant. Here, as x changes, y remains constant, being equal to c for all values of x. Hence, when x is given an increment Δx, the corresponding increment in yhas the value zero; or, Δy = 0. Dividing by Δx:

images = 0;

and since the difference-quotient is always zero, its limit is likewise zero. Therefore


In other words:

images(c) = 0.[1]

RULE. The derivative of a constant equals zero.


Geometrically, if the line RS represents the equation y = c, we get, by differentiating:


But images represents the slope of the line RS, or the function y = c; and since the line RS is parallel to OX, its slope equals zero. Therefore, as above,

images (c) = 0.

3—2. The Derivative of a Variable with Respect to itself. Consider the simple function, y = x. Applying the General Rule:

Step 1.y + Δy = x + Δx.

Step 2.Δy = Δx.

Step 3.images

Step 4.images

orimages (x) = 1.

RULE. The derivative of a variable with respect to itself equals unity.


Geometrically interpreted, the slope of the line y = x is constant and equal to unity at all points along the line; hence for all values of x,

images = 1, or images (x) = 1.

3—3. The Derivative of the Algebraic Sum of Two or More Functions. Consider the function

y = u + vw,

where u, v, and w are functions of x. Applying the General Rule, we have:

Step 1.y + Δy = u + Δu + v + Δvw − Δw.

Step 2.Δy = Δu + Δv − Δw.

Step 3.images.

Now, as Δx → 0, Δu → 0, Δv → 0, and Δw → 0; hence

Step 4.images


RULE. The derivative of the algebraic sum of any finite number of functions is equal to the algebraic sum of their individual derivatives.

3—4. The Derivative of the Product of a Constant and a Variable. Let the function be y = cv. By the General Rule:

Step 1.y + Δy = c(v + Δy) = cv + cΔv.

Step 2.Δy = cΔv.

Step 3.images

Now, as Δx → 0, Δv → 0; hence

Step 4.images


RULE. The derivative of the product of a constant and a variable is equal to the product of the constant and the derivative of the variable.

If the constant appears in the denominator, we have:


EXAMPLE 1.Find the derivative of x + 5.

Solution.images (x + 5) = images (x) + images (5) = 1 + 0 = 1.

EXAMPLE 2.Differentiate y = 3x + 2.

Solution.images = images (3x) + images (2) = 3 + 0 = 3.

EXAMPLE 3.Differentiate y = images + 5x − 1.



EXAMPLE 4.Differentiate y = 4(3x − 7).


Here c = 4, and v = 3x − 7

Therefore images = (4) images (3x − 7) = (4)(3) = 12.