## The Calculus Primer (2011)

### Part III. Differentiation of Algebraic Functions

### Chapter 8. THE DERIVATIVE OF A CONSTANT, A VARIABLE, AND A SUM

**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 Δ*x*, the corresponding increment in *y*has the value zero; or, Δ*y* = 0. Dividing by Δ*x*:

= 0;

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

In other words:

(*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 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,

(*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* + Δ*y* = *x* + Δ*x*.

*Step* 2.Δ*y* = Δ*x*.

*Step* 3.

*Step* 4.

or (*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*,

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

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

*y* = *u* + *v* − *w*,

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

*Step* 1.*y* + Δ*y* = *u* + Δ*u* + *v* + Δ*v* − *w* − Δ*w*.

*Step* 2.Δ*y* = Δ*u* + Δ*v* − Δ*w*.

*Step* 3..

Now, as Δ*x* → 0, Δ*u* → 0, Δ*v* → 0, and Δ*w* → 0; hence

*Step* 4.

or

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* + Δ*y* = *c*(*v* + Δ*y*) = *cv* + *c*Δ*v*.

*Step* 2.Δ*y* = *c*Δ*v*.

*Step* 3.

Now, as Δ*x* → 0, Δ*v* → 0; hence

*Step* 4.

or

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*. (*x* + 5) = (*x*) + (5) = 1 + 0 = 1.

EXAMPLE 2.Differentiate *y* = 3*x* + 2.

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

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

*Solution*.

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

*Solution*.

Here *c* = 4, and *v* = 3*x* − 7

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