Differentiating Other Functions (Trig, ln & e etc)

How do I differentiate common functions?

These are the common results
- $\frac{d}{d x} (x^{n}) = n x^{n - 1}$
- $\frac{d}{d x} (e^{x}) = e^{x}$
- $\frac{d}{d x} (a^{x}) = a^{x} \ln a$ for $a > 0$
- $\frac{d}{d x} (\ln x) = \frac{1}{x}$
- $\frac{d}{d x} (\sin x) = \cos x$
- $\frac{d}{d x} (\cos x) = - \sin x$
- $\frac{d}{d x} (\tan x) = \sec^{2} x$
- $\frac{d}{d x} (\cot x) = - \cos {ec}^{2} x$
- $\frac{d}{d x} (\sec x) = \sec x \tan x$
- $\frac{d}{d x} (\cos ec x) = - \cos ec x \cot x$
- $\frac{d}{dx} (\tan^{- 1} x) = \frac{1}{1 + x^{2}}$

How do I differentiate exponentials and logarithms?

The two basic differentiation formulae are:

Diff Other Funct Illustr 1, AS & A Level Maths revision notes

From those basic formulae are derived these two additional formulae:

Diff Other Funct Illustr 2_forms, AS & A Level Maths revision notes

Diff Other Funct Illustr 2_derivs, AS & A Level Maths revision notes

And for exponentials more generally:

Diff Other Funct Illustr 3, AS & A Level Maths revision notes

This last formula can be derived from Formula 3 by using the chain rule

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

Examiner Tip

All of the above are in the formula book, make sure you know how to find them.
They will be particularly useful when working with identities and when integrating trigonometric functions.
Recognising the derivative of tan^-1x will be particularly useful for integrating some functions.

Worked example

Diff Other Funct Example, AS & A Level Maths revision notes

Differentiating Other Functions (Trig, ln & e etc) (CIE A Level Maths: Pure 3)

Revision Note

How do I differentiate common functions?

How do I differentiate exponentials and logarithms?

How do I differentiate the reciprocal trigonometric functions?

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1. Algebra & Functions

1.1 Modulus Functions

1.1.1 Modulus Functions - Sketching Graphs

1.1.2 Modulus Functions - Solving Equations

1.2 Polynomials

1.2.1 Polynomial Division

1.2.2 Factor & Remainder Theorem

1.2.3 Factorisation

1.2.4 Rational Expressions

1.2.5 Top Heavy Rational Expressions

1.3 Partial Fractions

1.3.1 Linear Denominators

1.3.2 Squared Linear Denominators

1.3.3 Quadratic Denominators

1.4 Further Modelling with Functions

1.4.1 Further Modelling with Functions

1.5 General Binomial Expansion

1.5.1 General Binomial Expansion

1.5.2 General Binomial Expansion - Subtleties

1.5.3 General Binomial Expansion - Multiple

1.5.4 Approximating values

2. Logs & Exponentials

2.1 Logarithmic & Exponential Function

2.1.1 Exponential Functions

2.1.2 Logarithmic Functions

2.1.3 "e"

2.1.4 Derivatives of Exponential Functions

2.2 Laws of Logarithms

2.2.1 Laws of Logarithms

2.2.2 Exponential Equations

2.3 Modelling with Logs & Exponentials

2.3.1 Exponential Growth & Decay

2.3.2 Using Exps & Logs in Modelling

2.3.3 Using Log Graphs in Modelling

3. Trigonometry

3.1 Reciprocal Trigonometric Functions

3.1.1 Reciprocal Trig Functions - Definitions

3.1.2 Reciprocal Trig Functions - Graphs

3.1.3 Trigonometry - Further Identities

3.2 Compound & Double Angle Formulae

3.2.1 Compound Angle Formulae

3.2.2 Double Angle Formulae

3.2.3 R addition formulae Rcos Rsin etc

3.3 Further Trigonometric Equations

3.3.1 Strategy for Further Trigonometric Equations

3.4 Trigonometric Proof

3.4.1 Trigonometric Proof

4. Differentiation

4.1 Further Differentiation

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

4.1.2 Product Rule

4.1.3 Quotient Rule

4.2 Implicit Differentiation

4.2.1 Implicit Differentiation

4.3 Differentiation of Parametric Equations

4.3.1 Parametric Equations - Basics

4.3.2 Parametric Equations - Eliminating the Parameter

4.3.3 Parametric Equations - Sketching Graphs

4.3.4 Parametric Differentiation

5. Integration

5.1 Further Integration

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

5.1.2 Integrating with Trigonometric Identities

5.1.3 f'(x)/f(x)

5.1.4 Substitution (Reverse Chain Rule)

5.1.5 Harder Substitution

5.1.6 Integration by Parts

5.1.7 Integration using Partial Fractions

5.1.8 Integration Decision Making

5.2 Differential Equations

5.2.1 General Solutions

5.2.2 Particular Solutions