7.1.5
Finding derivatives
Differentiation
Differentiation
The derivative of a function can be worked out through a process called differentiation.
![Illustrative background for Differentiation](https://image-v2.cdn.app.senecalearning.com/2018-09/48aebd73-4676-49b6-97f4-595e3345bb94/writing-write-,h_400,q_80,w_640.jpg)
![Illustrative background for Differentiation ?? "content](https://image-v2.cdn.app.senecalearning.com/2018-09/48aebd73-4676-49b6-97f4-595e3345bb94/writing-write-,h_400,q_80,w_640.jpg)
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 .
- If your function starts with f() =, then the derivative of the equation is f'().
- The exam may also ask you to find the rate of change, the gradient of the tangent or to differentiate a function.
![Illustrative background for General rule](https://image-v2.cdn.app.senecalearning.com/2018-09/3bf436b2-5bb6-43b1-bf1c-a381fc5a26c0/rules-rule-board-,h_400,q_80,w_640.jpg)
![Illustrative background for General rule ?? "content](https://image-v2.cdn.app.senecalearning.com/2018-09/3bf436b2-5bb6-43b1-bf1c-a381fc5a26c0/rules-rule-board-,h_400,q_80,w_640.jpg)
General rule
General rule
- To differentiate a whole equation/function, each term should be differentiated individually.
- The general rule for differentiating a term is:
- f() = → f'() = −1
![Illustrative background for Examples](https://image-v2.cdn.app.senecalearning.com/2019-04/4b70f9e1-b8db-46e1-90ae-fd4e4671bfed/tangent-curve%20(1),h_400,q_80,w_640.png)
![Illustrative background for Examples ?? "content](https://image-v2.cdn.app.senecalearning.com/2019-04/4b70f9e1-b8db-46e1-90ae-fd4e4671bfed/tangent-curve%20(1),h_400,q_80,w_640.png)
Examples
Examples
- f() =
→
f'() =
−1
- f() = 3 → f'() = 32
- f() = 23 → f'() = 62
- f() = 6 → f'() = 6 (remember f() = 6 is the same as 61)
- f() = 34 → f'() = 123
- f() = 9 → f'() = 0 (remember f() = 9 is the same as 90)
![Illustrative background for Example 1 with multiple terms](https://image-v2.cdn.app.senecalearning.com/2019-04/4b70f9e1-b8db-46e1-90ae-fd4e4671bfed/tangent-curve%20(1),h_400,q_80,w_640.png)
![Illustrative background for Example 1 with multiple terms ?? "content](https://image-v2.cdn.app.senecalearning.com/2019-04/4b70f9e1-b8db-46e1-90ae-fd4e4671bfed/tangent-curve%20(1),h_400,q_80,w_640.png)
Example 1 with multiple terms
Example 1 with multiple terms
- For each term in an expression, use f() =
→
f'() =
−1
- f() = 3 + 22 + 7
- f'() = 32 + 4 + 7
![Illustrative background for Example 1 with multiple terms](https://image-v2.cdn.app.senecalearning.com/2019-04/4b70f9e1-b8db-46e1-90ae-fd4e4671bfed/tangent-curve%20(1),h_400,q_80,w_640.png)
![Illustrative background for Example 1 with multiple terms ?? "content](https://image-v2.cdn.app.senecalearning.com/2019-04/4b70f9e1-b8db-46e1-90ae-fd4e4671bfed/tangent-curve%20(1),h_400,q_80,w_640.png)
Example 1 with multiple terms
Example 1 with multiple terms
- For each term in an expression, use f() =
→
f'() =
−1
- f() = 34 + 53 + 42
- f'() = 123 + 152 + 8
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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