Differentiating Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA)): Revision Note

Author

Paul

Last updated

5 June 2025

Differentiating Reciprocal Trigonometric Functions

What are the reciprocal trigonometric functions?

Secant, cosecant and cotangent and abbreviated and defined as

$\sec x = \frac{1}{\cos x}$ $cosec x = \frac{1}{\sin x}$ $\cot x = \frac{1}{\tan x}$

Remember that for calculus, angles need to be measured in radians
- $θ$ may be used instead of $x$
$cosec x$ is sometimes further abbreviated to $\csc x$

What are the derivatives of the reciprocal trigonometric functions?

$f (x) = \sec x$
- $f' (x) = \sec x \tan x$
$f (x) = cosec x$
- $f' (x) = - cosec x \cot x$
$f (x) = \cot x$
- $f' (x) = - {cosec}^{2} x$
These are given in the formula booklet

How do I show or prove the derivatives of the reciprocal trigonometric functions?

For $y = \sec x$
- Rewrite, $y = \frac{1}{\cos x}$
- Use quotient rule, $\frac{d y}{d x} = \frac{\cos x (0) - (1) (- \sin x)}{\cos^{2} x}$
- Rearrange, $\frac{d y}{d x} = \frac{\sin x}{\cos^{2} x}$
- Separate, $\frac{d y}{d x} = \frac{1}{\cos x} \times \frac{\sin x}{\cos x}$
- Rewrite, $\frac{d y}{d x} = \sec x \tan x$
Similarly, for $y = cosec x$
- $y = \frac{1}{\sin x}$
- $\frac{d y}{d x} = \frac{\sin x (0) - (1) \cos x}{\sin^{2} x}$
- $\frac{d y}{d x} = \frac{- \cos x}{\sin^{2} x}$
- $\frac{d y}{d x} = - \frac{1}{\sin x} \times \frac{\cos x}{\sin x}$
- $\frac{d y}{d x} = - cosec x \cot x$

What do the derivatives of reciprocal trig look like with a linear functions of x?

For linear functions of the form ax+b
- $f (x) = \sec (a x + b)$
  - $f' (x) = a \sec (a x + b) \tan (a x + b)$
- $f (x) = cosec (a x + b)$
  - $f' (x) = - a cosec (a x + b) \cot (a x + b)$
- $f (x) = \cot (a x + b)$
  - $f' (x) = - a {cosec}^{2} (a x + b)$
- These are not given in the formula booklet
  - they can be derived from chain rule
  - they are not essential to remember

Examiner Tips and Tricks

Even if you think you have remembered these derivatives, always use the formula booklet to double check
- those squares and negatives are easy to get muddled up!
Where two trig functions are involved in the derivative be careful with the angle multiple; $x, 2 x, 3 x$ , etc
- An example of a common mistake is differentiating $y = c o s e c 3 x$
  - $\frac{d y}{d x} = - 3 c o s e c x c o t 3 x$ instead of $\frac{d y}{d x} = - 3 c o s e c 3 x c o t 3 x$

Worked Example

Curve C has equation $y = 2 \cot (3 x - \frac{π}{8})$ .

a) Show that the derivative of $\cot x$ is $- {cosec}^{2} x$ .

b) Find $\frac{d y}{d x}$ for curve C.

c) Find the gradient of curve C at the point where $x = \frac{7 π}{24}$ .

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Test yourself Flashcards

Did this page help you?

Differentiating Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA)): Revision Note

Differentiating Reciprocal Trigonometric Functions

What are the reciprocal trigonometric functions?

What are the derivatives of the reciprocal trigonometric functions?

How do I show or prove the derivatives of the reciprocal trigonometric functions?

What do the derivatives of reciprocal trig look like with a linear functions of x?

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

Partial Fractions, Indices & Standard Form

Standard Form

Laws of Indices

Partial Fractions

Exponentials & Logs

Introduction to Logarithms

Laws of Logarithms

Solving Exponential Equations

Sequences & Series

Language of Sequences & Series

Sigma Notation

Arithmetic Sequences & Series

Geometric Sequences & Series

Applications of Sequences & Series

Compound Interest & Depreciation

Simple Proof & Reasoning

Proof by Deduction

Disproof by Counter Example

Proof by Induction & Contradiction

Proof by Induction

Proof by Contradiction

Binomial Theorem

Binomial Theorem

Binomial Coefficients & Pascal's Triangle

Extension of The Binomial Theorem

Permutations & Combinations

Counting Principles

Permutations

Combinations

Complex Numbers

Introduction to Complex Numbers

Operations with Complex Numbers

Modulus & Argument

Introduction to Argand Diagrams

Further Complex Numbers

Geometry of Complex Numbers

Modulus-Argument (Polar) Form

Exponential (Euler's) Form

Conversion between Forms of Complex Numbers

Complex Roots of Polynomials

De Moivre's Theorem

Proof of De Moivre's Theorem

Roots of Complex Numbers

Systems of Linear Equations

Systems of Linear Equations

Solving Systems Using Row Reduction

Number of Solutions to a System

2. Functions

Linear Functions & Graphs

Equations of a Straight Line

Parallel & Perpendicular Lines

Quadratic Functions & Graphs

Quadratic Functions

Factorising Quadratics

Completing the Square

Solving Quadratic Equations

Quadratic Inequalities

Discriminants

Functions Toolkit

Language of Functions

Composite Functions

Inverse Functions

Odd & Even Functions

Periodic Functions

Self-Inverse Functions

Graphing Functions & Their Key Features

Intersecting Graphs

Other Functions & Graphs

Exponential & Logarithmic Functions

Solving Equations Analytically