Differentiating Special Functions (DP IB Maths: AI HL)

Revision Note

Author

Paul

Last updated

19 July 2024

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Differentiating Trig Functions

How do I differentiate sin, cos and tan?

The derivative of $y = \sin x$ is $\frac{d y}{d x} = \cos x$
The derivative of $y = \cos x$ is $\frac{d y}{d x} = - \sin x$
The derivative of $y = \tan x$ is $\frac{d y}{d x} = \frac{1}{\cos^{2} x}$
- This result can be derived using quotient rule
All three of these derivatives are given in the formula booklet
For the linear function $a x + b$ , where $a$ and $b$ are constants,
- the derivative of $y = \sin (a x + b)$ is $\frac{d y}{d x} = a \cos (a x + b)$
- the derivative of $y = \cos (a x + b)$ is $\frac{d y}{d x} = - a \sin (a x + b)$
- the derivative of $y = \tan (a x + b)$ is $\frac{d y}{d x} = \frac{a}{\cos^{2} (a x + b)}$
For the general function $f (x)$ ,
- the derivative of $y = \sin (f (x))$ is $\frac{d y}{d x} = f^{'} (x) \cos (f (x))$
- the derivative of $y = \cos (f (x))$ is $\frac{d y}{d x} = - f' (x) \sin (f (x))$
- the derivative of $y = \tan (f (x))$ is $\frac{d y}{d x} = \frac{f' (x)}{\cos^{2} (f (x))}$
These last three results can be derived using the chain rule
For calculus with trigonometric functions angles must be measured in radians
- Ensure you know how to change the angle mode on your GDC

Examiner Tip

As soon as you see a question involving differentiation and trigonometry put your GDC into radians mode

Worked example

Find

f' (x)

for the functions

$f (x) = \sin x$
$f (x) = \cos (5 x + 1)$

5-2-1-ib-hl-ai-aa-extraaa-we1a-soltn

b) A curve has equation

y = \tan (6 x^{2} - \frac{π}{4})

Find the gradient of the tangent to the curve at the point where

x = \frac{\sqrt{π}}{2}

Give your answer as an exact value.

5-2-1-ib-hl-ai-aa-extraaa-ai-we1b-soltn

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Differentiating e^x & lnx

How do I differentiate exponentials and logarithms?

The derivative of $y = e^{x}$ is $\frac{d y}{d x} = e^{x}$ where $x \in ℝ$
The derivative of $y = \ln x$ is $\frac{d y}{d x} = \frac{1}{x}$ where $x > 0$
For the linear function $a x + b$ , where $a$ and $b$ are constants,
- the derivative of $y = e^{(a x + b)}$ is $\frac{d y}{d x} = a e^{(a x + b)}$
- the derivative of $y = \ln (a x + b)$ is $\frac{d y}{d x} = \frac{a}{(a x + b)}$
  - in the special case $b = 0$ , $\frac{d y}{d x} = \frac{1}{x}$ ( $a$ 's cancel)
For the general function $f (x)$ ,
- the derivative of $y = e^{f (x)}$ is $\frac{d y}{d x} = f' (x) e^{f (x)}$
- the derivative of $y = \ln (f (x))$ is $\frac{d y}{d x} = \frac{f' (x)}{f (x)}$
The last two sets of results can be derived using the chain rule

Examiner Tip

Remember to avoid the common mistakes:
- the derivative of $\ln k x$ with respect to $x$ is $\frac{1}{x}$ , NOT $\frac{k}{x}$
- the derivative of $e^{k x}$ with respect to $x$ is $k e^{k x}$ , NOT $k x e^{k x - 1}$

Worked example

A curve has the equation $y = e^{- 3 x + 1} + 2 \ln 5 x$ .

Find the gradient of the curve at the point where $x = 2$ gving your answer in the form $y = a + b e^{c}$ , where $a, b$ and $c$ are integers to be found.

5-2-1-ib-sl-aa-only-we2-soltn

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Previous:5.1.3 Modelling with DifferentiationNext:5.2.2 Techniques of Differentiation

Differentiating Special Functions (DP IB Maths: AI HL)

Revision Note

How do I differentiate sin, cos and tan?

How do I differentiate exponentials and logarithms?

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

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1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Approximation & Estimation

1.1.3 GDC: Solving Equations

1.2 Exponentials & Logs

1.2.1 Exponents

1.2.2 Logarithms

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.4 Financial Applications

1.4.1 Compound Interest & Depreciation

1.4.2 Amortisation & Annuities

1.5 Complex Numbers

1.5.1 Intro to Complex Numbers

1.5.2 Modulus & Argument

1.5.3 Introduction to Argand Diagrams

1.6 Further Complex Numbers

1.6.1 Geometry of Complex Numbers

1.6.2 Forms of Complex Numbers

1.6.3 Applications of Complex Numbers

1.7 Matrices

1.7.1 Introduction to Matrices

1.7.2 Operations with Matrices

1.7.3 Determinants & Inverses

1.7.4 Solving Systems of Linear Equations with Matrices

1.8 Eigenvalues & Eigenvectors

1.8.1 Eigenvalues & Eigenvectors

1.8.2 Applications of Matrices

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

2.4 Functions Toolkit

2.4.1 Composite & Inverse Functions

2.5 Transformations of Graphs

2.5.1 Translations of Graphs

2.5.2 Reflections of Graphs

2.5.3 Stretches of Graphs

2.5.4 Composite Transformations of Graphs

2.6 Further Modelling with Functions

2.6.1 Properties of Further Graphs

2.6.2 Natural Logarithmic Models

2.6.3 Logistic Models

2.6.4 Piecewise Models

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Radian Measure

3.1.3 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Further Trigonometry

3.4.1 The Unit Circle

3.4.2 Simple Identities