Derivatives: Definition, Rules, and Examples, Slides of Mathematics

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Chapter 3

The Derivative

Gradient of a Straight Line

Recall that, the gradient of a straight line in the x, y -plane is constant, and can be obtained from any two points on the line. So,

m = change in y change in x

∆y ∆x

y 2 − y 1 x 2 − x 1

Gradient of a Curve

The gradient of a curve at a point P is given by the gradient of the straight line touching the curve at P. This line is called the tangent of the curve.

Gradient of a Curve

The gradient of a curve y = f (x) at a point P(a, f (a)) can be estimated with the gradient of a line segment PQ if Q is close to P. So,

gradient at P ≈ gradient of PQ = f (a + ∆x) − f (a) ∆x

The closer Q is to P, the more accurate is the estimate. Eventually, as ∆x → 0, gradient of PQ → gradient at P.

Rate of Change

Let y = f (x). Then ∆ ∆yx is the average rate of change of y with respect to x over some interval [x 1 , x 2 ].

As ∆x approaches 0, ∆ ∆yx approaches lim∆x→ 0 ∆ ∆yx (if the limit exists), and is called the instantaneous rate of change of y with respect to x at x 1.

Definition

Definition Let f be a function defined on an interval containing a. The derivative of f at a, denoted by f ′(a), is

f ′(a) = lim ∆x→ 0

f (a + ∆x) − f (a) ∆x if the limit exists. Very often, the limit is written as

f ′(a) = (^) xlim→a

f (x) − f (a) x − a

If f ′(a) exists, we say f is differentiable at a. If y = f (x), f ′(a) can also be written as y ′(a), dydx |x=a etc..

Standard Result

Given f (x), it is common to consider the derivative as a function of x, i.e., f ′(x). As we go along, we shall introduce the derivatives of elementary functions.

Standard Result 1. f (x) = c, c constant ⇒ f ′(x) = 0.

Standard Result

Given f (x), it is common to consider the derivative as a function of x, i.e., f ′(x). As we go along, we shall introduce the derivatives of elementary functions.

Standard Result 1. f (x) = c, c constant ⇒ f ′(x) = 0. Proof.

f ′(x) = lim ∆x→ 0

f (x + ∆x) − f (x) ∆x = lim ∆x→ 0

c − c ∆x

Standard Result

Standard Result 2. f (x) = xn, n integer ⇒ f ′(x) = nxn−^1. Proof. We prove the case n > 0. The proof for n < 0 is similar.

f ′(x) = lim t→x

f (t) − f (x) t − x = lim t→x

tn^ − xn t − x = lim t→x

(t − x)(tn−^1 + tn−^2 x + tn−^3 x^2 + · · · + txn−^2 + xn−^1 ) t − x = lim t→x (tn−^1 + tn−^2 x + · · · + xn−^1 ) = nxn−^1

Example

Example Let y = f (x) = x^5. Find f ′( 1 ) and f ′( 2 ), and the equation of the tangent at x = 2.

Differentiability vs Continuity

Differentiability vs Continuity

Theorem If f is differentiable at a, then f is continuous at a.

Differentiability vs Continuity

The converse is NOT true!

Algebra of Differentiations

Theorem If f ′(x) and g ′(x) exist, then • (f ± g )′(x) = f ′(x) ± g ′(x)

• (cf )′(x) = cf ′(x) for any constant c.