What is the derivative of sin x?

cos x, with x measured in radians. The book proves it from the definition of the derivative using the limit of sin(h/2)/(h/2) = 1.

How are the derivatives of tan, cot, sec and csc found?

By writing each one in terms of sin x and cos x and using the quotient rule (or the product rule), as the book does for tan x.

What is the derivative of csc x?

−csc x cot x. The minus sign matters: the derivatives of the three 'co' functions (cos, cot, csc) all carry a minus sign.

Why must angles be in radians?

The key limit sin θ/θ → 1 as θ → 0 holds only when θ is measured in radians, so all the derivative formulas assume radians.

Which board sets my paper if I study in Gujrat?

Gujrat district is listed under the Board of Intermediate and Secondary Education, Gujranwala. PECTAA writes the books. Your board sets the paper and the dates, so follow its notices.

Derivatives of Trigonometric Functions: PECTAA Class 12 Mathematics, Punjab Board

You already know how to differentiate x³ or a fraction of polynomials. But what is the rate of change of sin x? One careful limit gives the answer, and after that the other five trigonometric derivatives follow from rules you have already learned.

Unit 2 of the PECTAA Mathematics 12 book is Further Differentiation. Unit 1 ended with applications of exponential and logarithmic functions (section 1.11). Unit 2 opens with a review of differentiation basics (section 2.1) and then the derivatives of trigonometric functions (section 2.2). This note is for FSc Part 2 (2nd year, Class 12) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.

What to know

Review (section 2.1). The derivative of y = f(x) is

f′(x) = lim (h → 0) [f(x + h) − f(x)] / h, if the limit exists.

It gives the instantaneous rate of change of f at a point. The rules recalled in the book, with u and v differentiable functions of x and c a constant:

Rule	Statement
Power	d/dx (xⁿ) = n xⁿ⁻¹, n ∈ ℝ
Constant	d/dx (c) = 0
Constant multiple	d/dx (cu) = c u′
Sum and difference	d/dx (u ± v) = u′ ± v′
Product	(uv)′ = u′v + uv′
Quotient	(u/v)′ = (u′v − uv′)/v², v ≠ 0

Derivative of sin x from the definition (section 2.2). Angles are in radians. With f(x) = sin x:

f′(x) = lim (h → 0) [sin(x + h) − sin x] / h.
Use sin P − sin Q = 2 cos((P + Q)/2) sin((P − Q)/2): the top becomes 2 cos((2x + h)/2) sin(h/2).
So f′(x) = lim cos((2x + h)/2) × [sin(h/2)/(h/2)].
As h → 0, the first factor → cos x and the second → 1 (θ in radians).
Hence d/dx (sin x) = cos x.

The book leaves the matching result d/dx (cos x) = −sin x as a "Try yourself" proof by definition. It goes the same way using cos P − cos Q = −2 sin((P + Q)/2) sin((P − Q)/2).

The other four. Once sin and cos are known, write tan, cot, sec and csc in terms of them and use the quotient or product rule. The book's summary, with the correct sign on (vi):


(i) d/dx (sin x) = cos x	(ii) d/dx (cos x) = −sin x
(iii) d/dx (tan x) = sec² x	(iv) d/dx (cot x) = −csc² x
(v) d/dx (sec x) = sec x tan x	(vi) d/dx (csc x) = −csc x cot x

A memory aid: each "co" function (cos, cot, csc) has a minus sign in its derivative. The pairs also match up: tan with sec², cot with csc², sec with sec tan, csc with csc cot.

Key points

Radians are essential: sin θ/θ → 1 only in radians.
sin → cos, cos → −sin.
The rest come from the quotient rule.
Simplify answers with sin² x + cos² x = 1.

Worked examples

Example 1 (book Example 1). Find f′(x) if f(x) = tan x.

tan x = sin x / cos x.
Quotient rule: f′(x) = [cos x · cos x − sin x · (−sin x)] / cos² x.
= (cos² x + sin² x)/cos² x = 1/cos² x = sec² x.

Example 2 (book Example 2). Find dy/dx if y = x² csc x.

Product rule: dy/dx = (x²)′ csc x + x² (csc x)′.
= 2x csc x + x²(−csc x cot x).
= x csc x (2 − x cot x).

Example 3 (cot x). Show d/dx (cot x) = −csc² x.

cot x = cos x / sin x.
Derivative = [(−sin x) sin x − cos x cos x] / sin² x = −(sin² x + cos² x)/sin² x.
= −1/sin² x = −csc² x.

Example 4 (sec x). Show d/dx (sec x) = sec x tan x.

sec x = 1/cos x.
Derivative = [0 · cos x − 1 · (−sin x)] / cos² x = sin x / cos² x.
= (1/cos x)(sin x/cos x) = sec x tan x.

Common mistakes in the exam

Dropping the minus sign for cos, cot and csc.
Writing sec x instead of sec² x for the derivative of tan x.
Mixing up the order in the quotient rule numerator: it is u′v − uv′.
Forgetting the product rule in terms like x² csc x.
Using degrees in a limit argument.

Exam technique

In a proof by definition, name each identity and limit you use.
Write the rule you apply (product or quotient) before substituting.
Factor the final answer, as in x csc x (2 − x cot x).
Check your board's notice for the sections in your paper.

Practice questions

Differentiate y = x sin x. Answer: sin x + x cos x.
Differentiate y = 3 tan x − 2 cos x. Answer: 3 sec² x + 2 sin x.
Find d/dx (sin x / x). Answer: (x cos x − sin x)/x².
Differentiate y = csc x. Answer: −csc x cot x.
Find the slope of y = sin x at x = 0. Answer: cos 0 = 1.

Ready to practise this? Continue in 2nd Year Mathematics.