Derivative of Trigonometry & Inverse Trigonometry, Composite Functions | Solved Examples, FAQs

Written by Neeraj Anand | ANAND CLASSES
Published by ANAND TECHNICAL PUBLISHERS

🔍 Introduction

In calculus, derivatives represent the rate of change of a function with respect to a variable. For trigonometric functions, derivatives help us analyze how angles and their trigonometric values change with respect to variables like x. Understanding these derivatives is essential for solving problems in both CBSE board exams and JEE Mains & Advanced.

Differentiation Rule For Trigonometric Functions

The differentiation of six basic trigonometric functions are as follows:

Function	Derivative of Function
sin x	cos x
cos x	-sin x
tan x	sec2 x
cosec x	-cosec x cot x
sec x	sec x tan x
cot x	-cosec2 x

Proof of Derivative of Trigonometric Functions Formula

Differentiation of sin(x)

By the first principle of differentiation

(d/dx) sin x = lim _h→0 [{sin (x + h) – sin x} / {(x + h) – x}]

⇒ (d/dx) sin x = lim _h→0 [{sin x cos h + sin h cos x – sin x} / h]

⇒ (d/dx) sin x = lim _h→0 [{((cos h – 1) / h) sin x} + {(sin h / h) cos x}]

⇒ (d/dx) sin x = lim _h→0 [{(cos h – 1) / h} sin x] + lim _h→0 [(sin h / h) cos x]

⇒ (d/dx) sin x = 0.sin x + 1.cos x

⇒ (d/dx) sin x = cos x

Therefore, differentiation of sin x is cos x. 

Differentiation of cos(x)

By the first principle of differentiation

(d/dx) cos x = lim _h→0 [{cos (x + h) – cos x} / {(x + h) – x}]

⇒ (d/dx) cos x = lim _h→0 [{cos x cos h – sin h sin x – cos x} / h]

⇒ (d/dx) cos x = lim _h→0 [{((cos h – 1) / h) cos x} – {(sin h / h) sin x}]

⇒ (d/dx) cos x = lim _h→0 [{(cos h – 1) / h} cos x] – lim _h→0 [(sin h / h) sin x]

⇒ (d/dx) cos x = 0.cos x – 1.sin x

⇒ (d/dx) cos x = -sin x

Therefore, differentiation of cos x is -sin x.

Differentiation of tan(x)

tan x = sinx / cos x

⇒ (d/dx) tan x = (d/dx)[sinx / cos x]

By using quotient rule

(d/dx) tan x = [{(d/dx)sinx} cosx – {(d/dx) cos x} sinx] / cos²x

⇒ (d/dx) tan x = [cos x cos x – (-sin x) sin x] / cos²x [By 4 and 5]

⇒ (d/dx) tan x = [cos²x + sin²x] / cos²x

⇒ (d/dx) tan x = 1 / cos²x [By 3]

⇒ (d/dx) tan x = sec²x

Therefore, differentiation of tan x is sec² x.

Differentiation of cosec(x)

(d/dx) cosec x = (d/dx) [1 / sin x]

Using chain rule

(d/dx) cosec x = [-1 / sin²x] (d/dx) sin x

⇒ (d/dx) cosec x = [-1 / sin²x] cos x

⇒ (d/dx) cosec x = -[1 / sinx] [cos x / sin x]

⇒ (d/dx) cosec x = – cosec x cot x

Therefore, the differentiation of cosec x is – cosec x cot x.

Differentiation of sec(x)

(d/dx) sec x = (d/dx) [1 / cos x] [By 2]

Using chain rule

(d/dx) sec x = [-1 / cos²x] (d/dx) cos x

⇒ (d/dx) sec x = [-1 / cos²x] (-sin x)

⇒ (d/dx) sec x = [1 / cos x] [sin x / cos x]

⇒ (d/dx) sec x = sec x tan x

Therefore, the differentiation of sec x is sec x tan x.

Differentiation of cot(x)

(d/dx) cot x = (d/dx)[cosx / sin x]

By using quotient rule

(d/dx) cot x = [{(d/dx)cosx} sin x – {(d/dx) sin x} cos x] / sin²x

⇒ (d/dx) cot x = [(-sinx) sin x – (cosx) cos x] / sin²x [By 4 and 5]

⇒ (d/dx) cot x = [ -sin²x – cos² x] / sin²x

⇒ (d/dx) cot x = -[ sin²x + cos²x] / sin²x

⇒ (d/dx) cot x = -1 / sin²x

⇒ (d/dx) cot x = -cosec²x

Therefore, differentiation of cot x is -cosec² x.

Chain Rule and Trigonometric Function

The chain rule states that if p(q(x)) is a function then, the derivative of this function is given by the product of the derivative of p(q(x)) and derivative of q(x). The chain rule is used to differentiate composite functions. The chain rule is mostly used to differentiate the composite trigonometric functions easily.

Example: Find the derivative of f(x) = tan 4x

Solution:

f(x) = tan 4x

⇒ f'(x) = (d/dx) [tan 4x]

By applying chain rule

f'(x) = (d/dx) [tan 4x](d/dx)[4x]

⇒ f'(x) = (sec² 4x)(4)

Differentiation of Composite Trigonometric Function

To evaluate the differentiation of the composite trig functions we apply chain rule of differentiation. The composite trig functions are the functions in which the angle of the trigonometric function is itself a function. The differentiation of composite trigonometric functions can be easily evaluated by applying the chain rule and the differentiation formulas for trig functions.

Example: Find the derivative of f(x) = cos(x² +4)

Solution:

f(x) = cos(x² +4)

⇒ f'(x) = (d/dx) cos(x² +4)

By applying chain rule

f'(x) = (d/dx) [cos(x² +4)](d/dx)[x² +4]

⇒ f'(x) = -(2x)sin(x² +4)

Differentiation of Inverse Trigonometric Functions

The derivatives of six inverse trigonometric functions are as follows:

Function	Derivative of Function
sin-1 x	1/√(1 – x2)
cos-1 x	-1/√(1 – x2)
tan-1 x	1/(1 + x2)
cosec-1 x	1/[|x|√(x2 – 1)]
sec-1 x	-1/[|x|√(x2 – 1)]
cot-1 x	-1/(1 + x2)

Example: Find the derivative of f(x) = 3sin^-1x + 4cos^-1x

Solution:

f'(x) = (d/dx) [3sin^-1x + 4cos^-1x]

⇒ f'(x) = (d/dx) [3sin^-1x ]+ (d/dx) [4cos^-1x]

⇒ f'(x) = 3(d/dx) [sin^-1x ]+ 4(d/dx) [cos^-1x]

⇒ f'(x) = 3[1 / √(1 – x²)] + 4[-1 / √(1 – x²)]

⇒ f'(x) = 3[1 / √(1 – x²)] – 4[1 / √(1 – x²)]

⇒ f'(x) = [1 / √(1 – x²)] (3- 4)

⇒ f'(x) = -[1 / √(1 – x²)]

Solved Examples on Differentiation of Trigonometry Functions

Problem 1: Find the derivative of f(x) = tan 2x.

Solution:

f(x) = tan 2x

⇒ f'(x) = (d/dx) tan 2x

By applying chain rule

f'(x) = (d/dx) [tan 2x](d/dx)[2x]

⇒ f'(x) = (sec²2x)(2)

⇒ f'(x) = 2sec²2x

Problem 2: Find the derivative of y = cos x / (4x²)

Solution:

y = cos x / (4x²)

Applying quotient rule

y’ = [(d/dx)cosx(4x²) – cosx (d/dx)(4x²)] / (4x²)²

⇒ y’ = [(-sinx)(4x²) – cosx (8x)] / (16x⁴)

⇒ y’ = [-4x²sinx – 8xcosx] / (16x⁴)

⇒ y’ = [-4x(xsinx + 2cosx)] / (16x⁴)

⇒ y’ = – (x sinx + 2cosx) / (4x³)

Problem 3: Evaluate the derivative f(x) = cosec x + x tan x

Solution:

f(x) = cosec x + x tan x

By applying formula and product rule

f'(x) = (d/dx) cosec x + (d /dx) [x tan x]

⇒ f'(x) = -cosec x cot x + (d /dx) x (tan x) + x (d /dx) (tan x)

⇒ f'(x) = -cosec x cot x + tan x + xsec²x

Problem 4: Find the derivative of the function f(x) = 6x⁴cos x

Solution:

f(x) = 6x⁴cos x

By applying product rule

f'(x) = (d/dx) [6x⁴cos x]

⇒ f'(x) = 6[(d/dx) (x⁴)(cos x) + (x⁴) (d/dx)(cos x)]

⇒ f'(x) = 6[ 4x³cos x + x⁴(-sin x)]

⇒ f'(x) = 6[ 4x³cos x – x⁴sin x]

⇒ f'(x) = 6x³[ 4cos x – x sin x]

Problem 5: Evaluate the derivative: f(x) = (x + cos x) (1 – sin x)

Solution:

f(x) = (x + cos x) (1 – sin x)

By applying product rule

f'(x) = (d /dx) [(x + cos x) (1 – sin x)]

⇒ f'(x) = [(d /dx) (x + cos x)] (1 – sin x) + (x + cos x) [(d /dx) (1 – sin x)]

⇒ f'(x) = [(1 – sin x) (1 – sin x)] + [(x + cos x) (0 – cos x)]

⇒ f'(x) = (1 – sin x)² – (x + cos x) cos x

⇒ f'(x) = 1 + sin²x – 2 sinx – x cosx – cos²x

Practice Problems – Differentiation of Trigonometric Functions

Problem 1: Find the derivative of y = sin(x) + cos(x).

Problem 2: Calculate the derivative of y = 2sin(x) – 3cos(x).

Problem 3: Find the derivative of y = 2sin(3x).

Problem 4: Determine the derivative of y = tan(5x).

Problem 5: Find the derivative of y = sin(x) cos(x).

Problem 6: Calculate the derivative of f(x)=sin(x)+cos(x).

Problem 7: Determine the derivative of y = tan²(x).

Problem 8: Determine the derivative of f(x)=arcsin(x)+arctan(x).

Problem 9: Find the derivative of the function: f(x)=sin(cos(x²)).

Problem 10: Differentiate the following function using the product rule: f(x)=x⋅cos⁡(x)

Differentiation of Trigonometric Functions – FAQs

What is Differentiation?

Differentiation is a mathematical operation that calculates the rate at which a function changes with respect to its independent variable.

What is Trigonometric Function?

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of its sides. 

What are the Formulas for Differentiation of Trigonometric Functions?

The formula for the differentiation of trigonometric functions are:

• (d/dx) sin x = cos x
• (d/dx) cos x = -sin x
• (d/dx) tan x = sec² x
• (d/dx) cosec x = -cosec x cot x
• (d/dx) sec x = sec x tan x
• (d/dx) cot x = -cosec² x

What Methods are Used to Derive the Differentiation of Trigonometric Functions?

The different ways in which the differentiation of trigonometric functions formula can be derived are:

• By using the First Principle of the Derivatives
• By using the Quotient Rule
• By using the Chain Rule

What are common mistakes to avoid in trigonometric differentiation?

Common mistakes include forgetting to apply the chain rule, incorrectly differentiating trigonometric functions, or neglecting to apply the product or quotient rules when necessary. It’s also important to watch for sign errors, especially with functions like cos⁡(x) and tan⁡(x)

🧮 Applications of Derivatives in Trigonometry

1. Finding Slopes of Tangents: Derivatives help calculate the slope of a tangent at any given point on a trigonometric curve.
2. Maxima and Minima: Derivatives determine the highest and lowest points of sine, cosine, and tangent curves.
3. Solving Real-Life Problems: These derivatives model periodic phenomena like sound waves, light waves, and seasonal changes.

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📋 Important Tips for Students

• Memorize basic trigonometric derivatives for quick problem-solving in exams.
• Practice applying product, quotient, and chain rules on trigonometric functions.
• Use graphs to visualize how derivatives affect sine, cosine, and tangent functions.

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