## The Calculus Primer (2011)

### Part II. The Derivative of a Function

### Chapter 7. DIFFERENTIATION: FINDING THE DERIVATIVE

**2—7. The Process of Differentiation.** The operation of finding the derivative of a function is called *differentiation*. The derivative, as we have already seen, is written . This symbol is read: “the derivative of *y* with respect to *x*,” or, sometimes, the “*x*-derivative of *y*” It may be written in a variety of ways, but they all denote the same thing: thus

or*f*(*x*) = *f*′(*x*).

We shall now illustrate the operation of differentiation, and then formulate a General Rule of procedure.

EXAMPLE 1.Differentiate: *y* = 3*x* − 2.

*Solution*.

*y* = 3*x* − 2.(1)

Let *x* take on an increment Δ*x*, and let the corresponding increment in y be Δ*y*.

Then*y* + Δ*y* = 3(*x* + Δ*x*) − 2,

or*y* + Δ*y* = 3*x* + 3Δ*x* − 2.(2)

Subtract equation (1) from equation (2):

*y* + Δ*y* = 3*x* + 3Δ*x* − 2

*y*= 3*x* − 2

Δ*y* = 3Δ*x*(3)

Now divide equation (3) by Δ*x*:

Finally, let Δ*x* → 0:

= 3,

or = 3.

EXAMPLE 2.Differentiate: *y* = *x*^{2} + 5.

*Solution*.

*y* = *x*^{2} + 5.(1)

*y* + Δ*y* = (*x* + Δ*x*)^{2} + 5,

or*y* + Δ*y* = *x*^{2} + 2*x*·Δ*x*+ (Δ*x*)^{2} + 5.(2)

Subtracting (1) from (2):

*y* + Δ*y* = *x*^{2} + 2*x*·Δ*x* + (Δ*x*)^{2} + 5

*y*= *x*^{2}+ 5

Δ*y* = 2*x*·Δ*x* + (Δ*x*)^{2}(3)

Dividing (3) by Δ*x*:

= 2*x* + Δ*x*.

Let Δ*x* → 0:

= 2*x*,

or = 2*x*.

EXAMPLE 3.Differentiate: *y* = *x*^{2} + 3*x* + 8.

*Solution*.

*y* + Δ*y* = (*x* + Δ*x*)^{2} + 3(*x* + Δ*x*) + 8

*y* + Δ*y* = *x*^{2} + 2*x*·Δ*x* + (Δ*x*)^{2} + 3*x* + 3Δ*x* + 8

*y*= *x*^{2}+ 3*x*+ 8

Δ*y* = 2*x*·Δ*x* + (Δ*x*)^{2} + 3Δ*x*

Dividing through by Δ*x*:

= 2*x* + Δ*x* + 3.

Let Δ*x* → 0:

= 2*x* + 3.

or = 2*x* + 3.

EXAMPLE 4.Differentiate: *y* = *x*^{3}.

*Solution*.*y* + Δ*y* = (*x* + Δ*x*)^{3}.

*y* + Δ*y* = *x*^{3} + 3*x*^{2}·Δ*x* + 3*x*·(Δ*x*)^{2} + (Δ*x*)^{3}.

Subtracting:Δ*y* = 3*x*^{2}·Δ*x* + 3*x* (Δ*x*)^{2} + (Δ*x*)^{3}.

Dividing: = 3*x*^{2} + 3*x*(Δ*x*) + (Δ*x*)^{2}.

Passing to the limit:

= 3*x*^{2},

for, as Δ*x* → 0, the terms 3*x*(Δ*x*) and (Δ*x*)^{2} vanish.

EXAMPLE 5.Differentiate: *y* = .

**2—8. The General Rule for Differentiation.** We may now formulate the “four-step” rule for differentiating a function, as follows:

*Step* 1. Substitute (*x* + Δ*x*) for *x* in the given equation, thus giving *y* a new corresponding value, (*y* + Δ*y*).

*Step* 2. Subtract the given equation from the equation obtained in Step 1, thus obtaining an expression for Δ*y*.

*Step* 3. Divide the equation obtained in Step 2 by Δ*x*, thus obtaining a value for .

*Step* 4. Find the limit of as Δ*x* approaches zero as a limit. This limit is the required derivative.

**EXERCISE 2—1**

*Differentiate each of the following functions by the General Rule:*

**1.** *y* = 2*x* − 3

**2.** *y* = 4 − *x*^{2}

**3.** *y* = 2*x*^{2} + 3*x*

**4.** *y* = 10*x*^{2}

**5.** *y* = *x*^{2} − *x* + 6

**6.** *y* = 3*x*^{2} − 2*x* + 1

**7.** *y* = 2*x*^{3} + 3

**8.** *y* = *x*^{3} − 2*x*

**9.** *y* = (*x* + 1)(*x* − 2)

**10.** *y* = (2*x* − l)(3*x* + 2)

**11.***y* =

**12.***y* =

**13.***y* =

**14.***y* =

**15.***y* =

**16.***y* =

**2—9. Finding the Tangent to a Curve.** In the light of the discussion in §2—6, we are now able to find the slope of the tangent to a given curve at a given point, or the slope of a curve at any desired point, provided, of course, that the equation of the curve is given.

EXAMPLE 1.Find the slope of the curve *y* = *x*^{2} + 6 at the point where *x* = 5.

*Solution*.*y* = *x*^{2} + 6.

By the General Rule,

= 2*x* + Δ*x*,

and = 2*x*.

Hence, when *x* = 5, = 2(5) = 10. The slope of the tangent to *y* = *x*^{2} + 6 at the point where *x* = 5 is 10, as is the slope of the curve at that point. The inclination of the tangent is *ϕ* = arc tan 10, or approximately 84°17′.

EXAMPLE 2.Find the slope of the curve *y* = 3*x*^{2} − 4*x* + 8 at the point where *x* = 3.

*Solution*.By the General Rule,

= 6*x* − 4 + 3(Δ*x*),

and = 6*x* − 4.

Hence, when *x* = 3, = 6(3) − 4 = 14, the required slope.

**EXERCISE 2—2**

*By differentiating by the General Rule, find the slope of the tangent to each of the following curves at the point indicated:*

**1.** *y* = *x*^{2} − 5, where *x* = 6.

**2.** *y* = 4*x*^{2} + 3, where *x* = 1.

**3.** *y* = *x*^{2} − 3*x* + 6, where *x* = 4.

**4.** *y* = 3*x*^{2} + 4*x*, where *x* = − 2.

**5.** *y* = (*x* + 3)^{2}, where *x* = 0.

**6.** *y* = *x*^{3} − *x*^{2}, where *x* = −3.

**7.** *y* = *x*^{3} + 8, where *x* = −2.

**8.** *y* = *x*^{4} − 1, where *x* = .