Differential Coefficient

The differential coefficient of a function y = f(x) at x = a is f’(a). As far as the JEE exam is concerned, the differential coefficient is an important topic. Students can expect questions from this topic for any competitive exams. In this article, we will discuss the concept of differential coefficient and the conditions under which it exists.

Consider the function y = f(x). The differential coefficient of y at x = a is f’(a). f’(a) will exist only when LHD = RHD at x = a. When LHD = RHD at x = a, then f’(a) = f’(a^–) = f’(a⁺).

Let us consider a = 5. If we have to find out f’(5), then we can say that f’(5) exists only if f’(5^–) = f’(5⁺).

f’(5) = f’(5^–) = f’(5⁺)

f’(5) = lim_{h→ 0}[f(5+h) – f(5)]/h = lim_{h→ 0}[f(5-h) – f(5)]/-h

In general,

• f’(a⁺) = lim_{h→ 0}[f(a+h) – f(a)]/h = f’(a)
• f’(a^–) = lim_{h→ 0}[f(a-h) – f(a)]/-h = f’(a)
• f’(a) = lim_{x→ a}[f(x) – f(a)]/(x-a)

Differential Coefficient of Standard Functions

(1) d/dx (k) = 0, k is a constant

(2) d/dx (xⁿ) = nx^n-1

(3) d/dx (log x) = 1/x

(4) d/dx (e^x) = e^x

(5) d/dx (a^x) = a^x log_e a

(6) d/dx (sin x) = cos x

(7) d/dx (cos x) = -sin x

(8) d/dx (tan x) = sec² x

Also, Read:

Differential Equations

Differentiation of Functions

Solved Examples

Example 1: The differential coefficient of the given function

(1) cosec x

(2) tan x

(3) sec x

(4) cos x

Solution:

Let y = log_e (√((1 + sin x)/(1 – sin x))

Multiply the numerator and denominator by √(1 + sin x)

=> log_e (√((1 + sin x)²/(1 – sin² x))

= log_e (√((1 + sin x)²/cos² x)

= log_e ((1 + sin x)/cos x)

= log_e(sec x + tan x)

dy/dx = [1/(sec x + tan x)] × sec x tan x + sec²x

= [1/(sec x + tan x)] × sec x (tan x + sec x)

= sec x

Hence, option (3) is the answer.

Example 2: The differential coefficient of x⁶ with respect to x³ is

(1) 8x²

(2) 3x³

(3) 5x³

(4) 2x³

Solution:

Let u = x⁶

Differentiate with respect to x, we get

du/dx = 6x⁵

Let v = x³

dv/dx = 3x²

du/dv = (du/dx)/(dv/dx)

= 6x⁵/3x²

= 2x³

Hence, option (4) is the answer.

Example 3: Find the differential coefficient of the function x² log x sin x with respect to x.

(1) x² log x cos x + x sin x + 2x log x sin x

(2) x log x cos x + x sin x – 2x log x sin x

(3) x² log x cos x + x sin x + 2x² log x sin x

(4) x² log x cos x + x² sin x + 2x log x sin x

Solution:

Let y = x² log x sin x

dy/dx = x² d/dx (log x sin x) + 2x log x sin x

= x² log x cos x + x² sin x (1/x) + 2x log x sin x

= x² log x cos x + x sin x + 2x log x sin x

Hence, option (1) is the answer.

Example 4: If y = sin [ cos (sin x)], then dy/dx =

(1) -cos [cos (sin x)] sin (cos x) cos x

(2) -cos [cos (sin x)] sin (sin x) cos x

(3) cos [cos (sin x)] sin (cos x) cos x

(4) cos [cos (sin x)] sin (sin x) cos x

Solution:

Given y = sin [ cos (sin x)]

dy/dx = cos [ cos (sin x)] ×(d/dx) (cos sin x)

= cos [ cos (sin x)] ×(-sin (sin x) cos x

= – cos [cos (sin x)] sin (sin x) cos x

Hence, option (2) is the answer.

Example 5: Find the differential coefficient of sin 2nx/cos² nx.

(1) 2n cosec² nx

(2) 2n sec² nx

(3) 2n² sec² nx

(4) 2n² tan² nx

Solution:

Let y = sin 2nx/cos² nx

= 2 sin nx cos nx/cos² nx

= 2 sin nx/cos nx

= 2 tan nx

Differentiate with respect to x

dy/dx = 2 sec² nx . n

= 2n sec² nx

Hence, option (2) is the answer.

Video Lesson- Differential Coefficient