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# 2.2. Univariate Calculus: Definition of derivative

### Learning Objectives

You should be able to

• understand several ways to think about derivatives, including as instantaneous rate of change, the slope of a tangent line or as a linear approximation to a function near a given point.
• recognise and use a range of notation for derivatives
• write down the definition of a derivative as a limit of the slopes of secant lines and use this definition to compute derivatives of  simple examples
• recognise where a function might not be differentiable from features of its graph

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.

Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 2.2.1. Notions & notations of derivatives

How we can think of derivatives as instantaneous rates of change, slopes of tangent lines or lienar approximations.
$$\frac{\mathrm{dy} }{\mathrm{d} t}(a) = \left.\frac{\mathrm{d} }{\mathrm{d} t}(y) \right|_{t=a} = y'(a) = \dot{y}(a)$$

#### 2.2.2. Definition of a derivative

$$\frac{\mathrm{df} }{\mathrm{d} x}(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$

#### 2.2.3. Non-differentiability

See why points of discontinuity (sudden jumps), kinks and points with vertical tangents are all likely to be points of non-differentiability.

### Economic Application: Marginal concepts and elasticities

In Economics you will often encounter the word marginal and this will always mean a derivative. The following application considers three examples of marginal concepts: marginal cost, marginal revenue, and marginal propensity to consume. In addition, the quiz introduces the concept of elasticity - a notion that you will often encounter in Economics, which is related to, but distinct from, the notion of a derivative.

#### Economic Application Exercise

The first question in the quiz allows you to reflect on the mathematical and economic meaning of marginal cost, marginal revenue, and marginal propensity to consume. The second question introduces the concept of elasticity and asks you to reflect on how it is similar and different from the concept of a derivative.

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to If you want to test yourself further, then try the advanced quiz linked below.

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