Derivative

DERIVATIVE

Intuitively, a function is derivable or differentiable at a point c if the graph of the function in a neighbourhood of c is a smooth curve without sudden changes in the direction of the graph.

The central idea of differential calculus is the notion of derivative. The concept of derivative originated with the tangent at a point on curve. The solution to this problem also provided a way to calculate velocity and more generally the rate of change of a function. Calculus thus became the mathematics of change.

Definition 2.17 Derivative

Let f be a function defined in a neighbourhood (c - δ, c + δ) of a point c, where δ > 0 and small.

f is said to be derivable or differentiable at c if exists.

Then it is denoted by f'(c) and is called the derivative of f at c.

1. One-side Derivative

Since derivative is defined as a limit we may have derivative on one-side only the left derivative or the right derivative.

Right derivative at c is defined as f'(c +) = if the limit exists, h > 0.

Theorem 2.8 f'(c) exists if and only if f'(c-) and ƒ'(c+) exist and are equal.

Then f'(c)= f'(c−) = f'(c+)

Theorem 2.9 Differentiability ⇒ continuity

If f is differentiable at c, then f is continuous at c.

Proof: Given f is differentiable at c

Hence, f is continuous at c

But the converse is not true.

i.e., f is continuous at c does not imply ƒ is differentiable at c.

We shall show this in the next example 1.

2. Derivative as a Function

We have seen f'(c) as the derivative of f at the point c.

Suppose for a given function f, if the derivative ƒ'(x) exists for infinitely many points, x in its domain, then we can regard ƒ' as a function derived from f  by the limiting process The process of obtaining ƒ' from ƒ is called differentiation and ƒ' is called the first derivative of f. The domain of f' is the set {x | ƒ'(x) exists}.

If f'(x) exists at every point x in the domain of f, then we say ƒ is differentiable.

Obtaining f'(x) at an arbitrary point x is referred to as differentiation with respect to x.

The notation f' was introduced by Lagrange in late 18th century.

3. Leibnitz Notation

Leibnitz used y for f(x)

Δy for f(x + h) - f(x)

Δx for (x + h) - x = h

4. Differentiability in an Interval

Definition 2.18 A function ƒ defined in an open interval (a, b) is said to be differentiable in (a, b) if ƒ'(x) exists ∀x∈(a, b).

A function ƒ defined on a closed interval [a, b] is said to be differentiable on [a, b] if

(i) f'(x) exists ∀x∈(a, b)

(ii) ƒ'(a+) exists

and (iii) ƒ'(b−) exists.

We shall first apply the definition in some simple functions.

WORKED EXAMPLES

Example 1 

Test the function |x| differentiable at x = 0.

Solution 

Hence f'(0) does not exist i.e., f is not differentiable at x = 0

We have seen in Example 12, section 2.4, |x| is continuous at x = 0 [and in fact |x| is continuous ∀x∈ (-∞, ∞)].

So continuity does not imply differentiability.

* But not continuous ⇒ not differentiable

Example 2 

Find the derivative of the function f(x) = x2 − 8x + 9 at the point 2.

Solution 

Given f(x) = x2 − 8x + 9

Required f'(2)

Example 3 

If f(x) = x3 − x, find a formula for ƒ'(x)

Solution 

Given f(x) = x3 − x

Example 4 

Where the function f(x)= |x| is differentiable?

Solution 

Given

We have seen in example 1, f is not differentiable at x = 0

If x > 0,  |x| = x

For any c in (0, ∞)

Since c > 0, |c| = c

We can choose h very small so that c + h > 0

So, f'(c) exists and c is arbitrary in (0, ∞)

⸫ f is differentiable is (0, ∞)

Now, consider the interval (-∞, 0) and let c be any point in (-∞, 0)

5. Higher Order Derivatives

Given a differentiable function ƒ we have seen the process of getting its derivative f'.

If f' is also differentiable we can find its derivative (f')'= ƒ" and is called the second derivative of f.

Proceeding this way can be find the nth derivative ƒ(n)of f, which is the first derivative of f(n−1)

We shall now discuss the general techniques or rules of differentiation.

Sum Rule

3. If u and v are differentiable functions of x then u + v and u – v are differentiable.

4. Product Rule

If u and v are differentiable functions of x, then uv is differentiable and

Proof:

Let f(x) = u(x) v(x)

5. Quotient Rule

If u and v are differentiable functions of x and if v(x) ≠ 0, then

We shall now find derivatives of some standard functions.

6. Power Function

1. where n is a rational number.

Proof: Let f(x) = xn

7. Exponential Function

2. = (ax) loge a, a > 0, a ≠ 1 

Proof:

Let f(x) = ax

Then

3.

Proof:

Let f(x) = loge x

4.

Proof:

Let f(x) = sin x

5.

Proof:

Let f(x) = cos x

6.

Proof:

Let f(x) = tan x

Similarly, we can prove

* Any polynomial function is differentiable everywhere.

WORKED EXAMPLES

Example 5 

Solution 

Given f(x) =

To test f is differentiable at 1, we find the left and right derivatives at 1.

f(1) = 2 - 1 = 1

Hence, f'(1) does not exist.

So, f is not differential at x = 1

But in the open interval (-∞, 1), f(x) = 2 − x, which is differentiable

⸫ f'(x) = -1

In the open interval (1, ∞), f(x) = x2 − 2x + 2, which is a polynomial and so differentibale

f'(x) = 2x − 2

Example 6 

At what numbers, the following function g differentiable?

Give a formula for f'. Sketch the graph of ƒ and f'.

Solution 

Given f(x) =

We test for derivative at x = -1 and x = 1

At x = -1

⸫ ƒ'(1-) ≠ ƒ'(1+)

Hence, f'(1) does not exist

In the interval (-∞, −1), f(x) = -1 - 2x, which is a polynomial

⸫ f'(x) = -2

In the interval (−1, 1), f(x) = x2

⸫ f'(x) = 2x

In the interval (1, ∞), f'(x) = x

f'(x) = 1

We shall draw the graph of ƒ and ƒ'. 

If x = 1, y = -2x - 1, which is a straight line

It meets the x-axis at the point (-1/2, 0) and the y-axis at the point (0, -1)

If −1 ≤ x ≤ 1, then graph of y = x2 is a parabola upward. 

If x > 1, then graph of y = x is a straight line.

Example 7 

For what values of x, is the function f(x) = |x2 − 9| differentiable? Find a formula for ƒ'.

Solution 

We test for derivative at x = −3 and x = 3 

At x = -3

f(x) is defined for all x∈ (−∞, ∞), it is differentiable for all x ∈ (-∞, ∞) except x = -3 and x = 3

Example 8 

For what values of x, the function f(x) = |x −1| + |x + 2| is differentiable?

Give formula for ƒ' and sketch ƒ and ƒ'

Solution 

Given f(x) = |x −1| + |x + 2|

Mark the points x = -2 and x = 1

Then we have 3 intervals (-∞, −2), (−2, 1), (1, ∞)

If x < -2, then x + 2 < 0, x - 1 < 0

It can be easily seen f(x) is continuous for all x ∈ (−∞, ∞)

We test for differentiability at x = 1, x = -2

At x = -2:

f'(1) does not exist.

f(x) is differentiable for all x∈(-∞, ∞), except x = −2 and x = 1.

and

We shall now draw the graph of ƒ.

When x = 0, y = 1 and when y = 0, x = -1/2

Remark: Look at the graph of f, at x = -2 the graph changes from y = -2x - 1 to y = 3 and the changes is not smooth because ƒ' (−2) does not exist. At x = 1, the graph changes form y = 3 to y = 2x + 1, and the change is not smooth because ƒ' (1) does not exist.

Example 9 

Find ƒ'(1), if it exists.

Solution

Given

Example 10 

f(x) =

Prove that f is continuous but not deliverable at x = 0.

Solution

Hence, f is continuous at x = 0

which does not exist

⸫ f is not differentiable at x = 0, but differentiable everywhere else

Example 11 

Show that f(x) =

Is differentiable at x = 0 and find ƒ'(0).

Solution 

So, f is differentiable at x = 0 and ƒ'(0) = 0

Example 12 

Find the values of a and b so that f is differentiable ∀x.

Solution 

Given

Suppose ƒ is differentiable for all x, then f is differentiable at x = 2

Hence, f is continuous at x = 2

Substitute in (1),

b = 4 – 2a = 4 – 8 = -4

a = 4, b = -4

Note: A useful result for computation of problems of above type.

If f is continuous at c and if f'(x) exists, then f'(x) = ƒ'(c).

Example 13 

For the function f(x) :

Find ƒ'(0+) and ƒ'(0–).

Solution 

Given

EXERCISE 2.5

1.

At what points fis differentiable and give a formula for ƒ'.

2. Find the points where f(x) = |4 - x2| is not differentiable. Find a formula for f'.

3. Show that f(x) = is continuous and differentiable at x = 1

4. Show that f(x) = is continuous but not differentiable at x = 1

5. f(x) =

find a and b if ƒ'(2) exists.

 [Hint: f is continuous at x = 2]

6. Does the function f(x) = have a derivative at x = 0? Explain.

7. For what values of a and b will f(x) = differentiable for all values of x.

8. For what values of a and b will g(x) = is differentiable at x = -1

9. f(x) = Does f'(0) exist?

10. Show that the function f(x) = |x − 6| is not differentiable at x = 6. Find a formula for f'.

Using the definition of derivative (or first principles) find the derivative of the following functions.

ANSWERS TO EXERCISE 2.5