Differentiating Reciprocal and Inverse Trig Functions

How do I differentiate the reciprocal trigonometric functions?

The formulae for the derivatives of the reciprocal trigonometric functions are:

\frac{d}{d x} (\sec x) = \sec x \tan x

\frac{d}{d x} (cosec x) = - cosec x \cot x

\frac{d}{d x} (\cot x) = - {cosec}^{2} x

You can derive the derivatives for sec, cosec, and cot using the chain rule and the derivatives of the basic trigonometric functions

Diff Rec Inv Trig Illustr 1, AS & A Level Maths revision notes

How do I differentiate the inverse trigonometric functions?

The formulae for the derivatives of the inverse trigonometric functions are:

\frac{d}{d x} (\arcsin x) = \frac{1}{\sqrt{1 - x^{2}}}

\frac{d}{d x} (arc \cos x) = - \frac{1}{\sqrt{1 - x^{2}}}

\frac{d}{d x} (arc \tan x) = \frac{1}{1 + x^{2}}

You can derive the derivatives of arcsin, arccos, and arctan using the reciprocal form of the chain rule and the derivatives of the basic trigonometric functions

Examiner Tip

The formulae for the derivatives of sec x, cosec x, and cot x are in the formulae booklet – you don't need to memorise them
- However, you should know how to derive the derivatives of sec, cosec, and cot using the chain rule
The formulae for the derivatives of arcsin, arccos, and arctan are not in the formulae booklet
- You should know how to derive those derivatives using the reciprocal form of the chain rule

Worked example

Diff Rec Inv Trig Example, AS & A Level Maths revision notes

Differentiating Reciprocal and Inverse Trig Functions (Edexcel International A Level Maths: Pure 3)

Revision Note

How do I differentiate the reciprocal trigonometric functions?

How do I differentiate the inverse trigonometric functions?

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1. Algebra & Functions

1.1 Rational Expressions

1.1.1 Rational Expressions

1.1.2 Top Heavy Rational Expressions

1.2 Functions

1.2.1 Language of Functions

1.2.2 Composite Functions

1.2.3 Inverse Functions

1.2.4 Modulus Functions - Sketching Graphs

1.2.5 Modulus Functions - Solving Equations

1.3 Combinations of Transformations

1.3.1 Combinations of Transformations

2. Trigonometry

2.1 Reciprocal & Inverse Trigonometric Functions

2.1.1 Reciprocal Trig Functions - Definitions

2.1.2 Reciprocal Trig Functions - Graphs

2.1.3 Trigonometry - Further Identities

2.1.4 Inverse Trig Functions

2.2 Compound & Double Angle Formulae

2.2.1 Compound Angle Formulae

2.2.2 Double Angle Formulae

2.2.3 R addition formulae Rcos Rsin etc

2.2.4 Modelling with Trigonometric Functions

2.3 Further Trigonometric Equations

2.3.1 Strategy for Further Trigonometric Equations

2.4 Trigonometric Proof

2.4.1 Trigonometric Proof

3. Logs & Exponentials

3.1 Exponential & Logarithms

3.1.1 e & ln

3.1.2 Derivatives of Exponential Functions

3.2 Modelling with Exponentials & Logarithms

3.2.1 Exponential Growth & Decay

3.2.2 Using Exps & Logs in Modelling

3.2.3 Using Log Graphs in Modelling

4. Differentiation

4.1 Further Differentiation

4.1.1 Differentiating Other Functions (Trig, ln & e etc)

4.1.2 Chain Rule

4.1.3 Product Rule

4.1.4 Quotient Rule

4.1.5 Differentiating Reciprocal and Inverse Trig Functions

5. Integration

5.1 Further Integration

5.1.1 Integrating Other Functions (Trig, ln & e etc)

5.1.2 Reverse Chain Rule

5.1.3 f'(x)/f(x)

5.1.4 Integrating with Trigonometric Identities

6. Numerical Methods

6.1 Numerical Methods

6.1.1 Change of Sign

6.1.2 Change of Sign Failure

6.1.3 x = g(x) Iteration

Author: Roger