Derivatives of Functions

Derivatives

The derivative of a function, f(x), is the function that describes how the gradient changes as x changes. This is the same as saying that the derivative the rate of change of an equation.

Derivatives

The derivative of a function can be used to calculate the gradient of the function at any point.
The derivative of f( $x$ ) is called f'( $x$ ).
- f'( $x$ ) can be called "f dash of $x$ " or "f prime".

Example

A function, f( $x$ ), has the derivative, f'( $x$ ) = 2 $x$ + 5
What is the gradient of f( $x$ ) when $x$ = 3?
- Gradient at $x$ = 3 is f'(3).
- f'(3) = (2 × 3) + 5 = 11
f( $x$ ) has a gradient of 11 when $x$ = 3.

Differentiation

The derivative of a function can be worked out through a process called differentiation.

Differentiation

There are different ways to display the derivative of a function.
- If your equation starts with y =, then the derivative of the equation is $\large\frac{dy}{dx}$ .
- If your function starts with f( $x$ ) =, then the derivative of the equation is f'( $x$ ).
- The exam may also ask you to find the rate of change, the gradient of the tangent or to differentiate a function.

General rule

To differentiate a whole equation/function, each term should be differentiated individually.
The general rule for differentiating a term is:
- f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}

Examples

f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}
- f( $x$ ) = $x$ ³ → f'( $x$ ) = 3 $x$ ²
- f( $x$ ) = 2 $x$ ³ → f'( $x$ ) = 6 $x$ ²
- f( $x$ ) = 6 $x$ → f'( $x$ ) = 6 (remember f( $x$ ) = 6 $x$ is the same as 6 $x$ ¹)
- f( $x$ ) = 3 $x$ ⁴ → f'( $x$ ) = 12 $x$ ³
- f( $x$ ) = 9 → f'( $x$ ) = 0 (remember f( $x$ ) = 9 is the same as 9 $x$ ⁰)

Example 1 with multiple terms

For each term in an expression, use f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}
- f( $x$ ) = $x$ ³ + 2 $x$ ² + 7 $x$
- f'( $x$ ) = 3 $x$ ² + 4 $x$ + 7

Example 1 with multiple terms

For each term in an expression, use f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}
- f( $x$ ) = 3 $x$ ⁴ + 5 $x$ ³ + 4 $x$ ²
- f'( $x$ ) = 12 $x$ ³ + 15 $x$ ² + 8 $x$

Derivatives in Kinematics

Another use for the derivative is to analyse motion along a line.

Velocity

The velocity of an object in motion is the rate of change of its displacement.
So the derivative of an object's displacement as a function of time is equal to the velocity, or:
- v(t) = s'(t)
Where v(t) is the velocity and s(t) is the position, both as a function of time

Acceleration

Taking the derivative of the velocity gives the acceleration of the object, which is the rate of change of velocity.
- a(t) = v'(t) = s''(t)
So the acceleration can also be thought of as the second derivative of the object's displacement with respect to time.

1Numbers

1.1Integers

1.1.1Integers

1.1.2Ordering Numbers

1.1.3The Four Rules of Operations

1.1.4Hierarchy of Operations

1.1.5Definitions & Terms

1.1.6HCF & LCM

1.1.7Standard Form

1.1.8Standard Form 2

1.1.9Prime Numbers

1.1.10End of Topic Test - Standard Form

1.2Fractions

1.2.1Fractions

1.2.2Using Fractions

1.2.3Conversion

1.2.4Reciprocals

1.3Decimals

1.3.1Decimals

1.3.2Decimals 2

1.3.3Exam-Style Questions - Recurring Decimals

1.3.4Comparing & Converting Decimals

1.3.5End of Topic Test - Fractions & Decimals

1.4Powers & Roots

1.4.1Square Numbers

1.4.2Index Notation

1.4.3Exam-Style Questions - Index Notation

1.4.4Advanced Indices

1.4.5Surds

1.4.6End of Topic Test - Powers & Roots

1.5Set Language & Notation

1.5.1Sets

1.6Percentages

1.6.1Percentages

1.6.2Percentage Change

1.6.3Percentages in Finance

1.6.4Compound Interest

1.7Ratio & Proportion

1.7.1Manipulating Ratios

1.7.2Proportion

1.8Degree of Accuracy

1.8.1Rounding & Significant Figures

1.8.2Estimation

1.8.3Upper & Lower Bounds

1.8.4Limits of Accuracy

1.8.5Limits of Accuracy 2

1.8.6End of Topic Test - Rounding & Estimation

2Equations, Formulae & Identities

2.1Algebraic Manipulation

2.1.1Algebra Basics

2.1.2Power Rules & Collecting Terms

2.1.3Expanding Brackets

2.1.4Expanding Brackets 2

2.1.5Factorising Quadratics

2.1.6Algebraic Fractions

2.1.7Proofs

2.1.8End of Topic Test - Algebra & Proofs

2.1.9Exam-Style Questions - Algebraic Proof

2.2Expressions & Formulae

2.2.1Rearranging Formulae

2.2.2Simple Formulae

2.2.3End of Topic Test - Algebra Manipulation & Formula

2.3Linear Equations

2.3.1Solving Linear Equations

2.3.2Simultaneous Linear Equations

2.4Quadratic Equations

2.4.1Solving Quadratics

2.4.2Solving Quadratics 2

2.4.3End of Topic Test - Solving Equations

2.4.4Simulateous Quadratic Equations

2.4.5Exam-Style Questions - Simultaneous Equations

2.4.6Graphs of Simultaneous Equations

2.4.7End of Topic Test - Simultaneous Equations

2.5Proportion

2.5.1Graphs & Proportion

2.6Inequalities

2.6.1Solving Inequalities

2.6.2Graphs of Inequalities

2.6.3Exam-Style Questions - Inequalities

2.6.4End of Topic Test - Inequalities

3Sequences, Functions & Graphs

3.1Sequences

3.1.1Sequence Rules

3.1.2Finding Terms

3.1.3Exam-Style Questions - Sequences

3.1.4End of Topic Test - Sequences

3.2Functions

3.2.1Functions

3.2.2Composite & Inverse

3.2.3End of Topic Test - Functions

3.3Graphs

3.3.1Plotting Graphs

3.3.2Straight Line Graphs

3.3.3Properties of Graphs

3.3.4Quadratic Graphs

3.3.5Practical Graphs

3.3.6Graphs of Polynomials

3.3.7End of Topic Test Graphs

3.4Common Graphs

3.4.1Graphs of Common Functions

3.4.2Transformations of Graphs

3.4.3End of Topic Test - Transformations and Graph

3.5Calculus

3.5.1Derivatives of Functions

3.5.2Maxima & Minima

4Geometry

4.1Angles, Lines & Triangles

4.1.1Angles & Lines

4.1.2Angle Rules

4.1.3Angles in Triangles

4.1.4Exam-Style Questions - Angles

4.2Polygons

4.2.1Polygons

4.2.2Quadrilaterals

4.2.3Congruence

4.2.4Scale Factor

4.2.5Exam-Style Questions - Enlargements

4.2.6End of Topic Test - Geometry

4.3Symmetry

4.3.1Symmetry

4.3.2Exam-Style Questions - Reflection

4.4Measures

4.4.1Measurements

4.4.2Time

4.4.3Bearings

4.4.4Speed

4.5Circle Properties

4.5.1Properties of a Circle

4.5.2Circle Theorems

4.5.3Circle Theorems 2

4.5.4Circle Theorems 3

4.5.5Exam-Style Questions - Circle Theorems

4.6Trigonometry & Pythagoras’ theorem

4.6.1Pythagoras' Theorem

4.6.2Trigonometry

4.6.3Sine & Cosine for Obtuse Angles

4.6.4Triangle Formulae

4.6.5Exam-Style Questions - Trigonometry & Area

4.7Mensuration of 2D Shapes

4.7.1Standard Units

4.7.2Perimeter of 2D Shapes

4.7.3Area of 2D Shapes

4.7.4Area & Circumference of Circle

4.7.5End of Topic Test - Trigonometry & Shapes

4.83D Shapes & Volume

4.8.13D Shapes

4.8.2Surface Area

4.8.3Volume

4.8.4Exam-Style Questions - Volume

5Vectors & Transformation Geometry

5.1Vectors

5.1.1Vectors

5.1.2Vectors 2

5.1.3Exam-Style Questions - Vectors

5.2Transformation Geometry

5.2.1Transformations

5.2.2Rotations

5.2.3Translations & Reflections

5.2.4Enlargements

5.2.5Effects of Transformations

6Statistics & Probability

6.1Statistical Measures

6.1.1Basic Statistics

6.1.2Frequency Table

6.2Graphical Representation of Data

6.2.1Pictograms, Bar Charts & Pie Charts

6.2.2Histograms

6.2.3Cumulative Frequency

6.2.4Cumulative Frequency 2

6.2.5Exam-Style Questions - Cumulative Frequency

6.2.6End of Topic - Datasets

6.3Probability

6.3.1Probability

6.3.2Theoretical Probability

6.3.3Experimental Probability

6.3.4Venn Diagrams

6.3.5Sample Space Diagrams

6.3.6Tree Diagrams

6.3.7Independent Events

6.3.8Conditional Probability

6.3.9End of Topic Test - Probability

6.3.10Exam-Style Questions - Probability

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