# 2: Derivatives

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Calculus is built on two operations — differentiation, which is used to analyse instantaneous rate of change, and integration, which is used to analyse areas. Understanding differentiation and using it to compute derivatives of functions is one of the main aims of this course.

We had a glimpse of derivatives in the previous chapter on limits — in particular Sections 1.1 and 1.2 on tangents and velocities introduced derivatives in disguise. One of the main reasons that we teach limits is to understand derivatives. Fortunately, as we shall see, while one does need to understand limits in order to correctly understand derivatives, one does not need the full machinery of limits in order to * compute and work with* derivatives. The other main part of calculus, integration, we (mostly) leave until a later course.

The derivative finds many applications in many different areas of the sciences. Indeed the reason that calculus is taken by so many university students is so that they may then use the ideas both in subsequent mathematics courses and in other fields. In almost any field in which you study quantitative data you can find calculus lurking somewhere nearby.

Its development ** ^{1}** came about over a very long time, starting with the ancient Greek geometers. Indian, Persian and Arab mathematicians made significant contributions from around the \(6^{th}\) century. But modern calculus really starts with Newton and Leibniz in the \(17^{th}\) century who developed independently based on ideas of others including Descartes. Newton applied his work to many physical problems (including orbits of moons and planets) but didn't publish his work. When Leibniz subsequently published his “calculus”, Newton accused him of plagiarism — this caused a huge rift between British and continental-European mathematicians which wasn't closed for another century.

- 2.1: Revisiting Tangent Lines
- By way of motivation for the definition of the derivative, we return to the discussion of tangent lines that we started in the previous chapter on limits. We consider, in Examples 2.1.2 and 2.1.5, below, the problem of finding the slope of the tangent line to a curve at a point. But let us start by recalling, in Example 2.1.1, what is meant by the slope of a straight line.

- 2.2: Definition of the Derivative
- We now define the “derivative” explicitly, based on the limiting slope ideas of the previous section. Then we see how to compute some simple derivatives.

- 2.3: Interpretations of the Derivative
- In the previous sections we defined the derivative as the slope of a tangent line, using a particular limit. This allows us to compute “the slope of a curve” 1 and provides us with one interpretation of the derivative. However, the main importance of derivatives does not come from this application. Instead, (arguably) it comes from the interpretation of the derivative as the instantaneous rate of change of a quantity.

- 2.4: Arithmetic of Derivatives - a Differentiation Toolbox
- So far, we have evaluated derivatives only by applying Definition 2.2.1 to the function at hand and then computing the required limits directly. It is quite obvious that as the function being differentiated becomes even a little complicated, this procedure quickly becomes extremely unwieldy.

- 2.5: Proofs of the Arithmetic of Derivatives
- The theorems of the previous section are not too difficult to prove from the definition of the derivative (which we know) and the arithmetic of limits (which we also know). In this section we show how to construct these rules.

- 2.6: Using the Arithmetic of Derivatives – Examples
- In this section we illustrate the computation of derivatives using the arithmetic of derivatives — Theorems 2.4.2, 2.4.3 and 2.4.5. To make it clear which rules we are using during the examples we will note which theorem we are using:

- 2.7: Derivatives of Exponential Functions
- Now that we understand how derivatives interact with products and quotients, we are able to compute derivatives of polynomials, rational functions, and powers and roots of rational functions.

- 2.8: Derivatives of Trigonometric Functions
- We are now going to compute the derivatives of the various trigonometric functions, \(\sin x\text{,}\) \(\cos x\) and so on. The computations are more involved than the others that we have done so far and will take several steps. Fortunately, the final answers will be very simple.

- 2.9: One More Tool – the Chain Rule
- We have built up most of the tools that we need to express derivatives of complicated functions in terms of derivatives of simpler known functions. We started by learning how to evaluate derivatives of sums, products and quotients derivatives of constants and monomials

- 2.10: The Natural Logarithm
- The chain rule opens the way to understanding derivatives of more complicated function. Not only compositions of known functions as we have seen the examples of the previous section, but also functions which are defined implicitly.

- 2.11: Implicit Differentiation
- Implicit differentiation is a simple trick that is used to compute derivatives of functions either when you don't know an explicit formula for the function, but you know an equation that the function obeys or even when you have an explicit, but complicated, formula for the function, and the function obeys a simple equation.

- 2.12: Inverse Trigonometric Functions
- One very useful application of implicit differentiation is to find the derivatives of inverse functions. We have already used this approach to find the derivative of the inverse of the exponential function — the logarithm.

- 2.13: The Mean Value Theorem
- Consider the following situation. Two towns are separated by a 120km long stretch of road.

- 2.14: Higher Order Derivatives
- The operation of differentiation takes as input one function, \(f(x)\text{,}\) and produces as output another function, \(f'(x)\text{.}\) Now \(f'(x)\) is once again a function. So we can differentiate it again, assuming that it is differentiable, to create a third function, called the second derivative of \(f\text{.}\) And we can differentiate the second derivative again to create a fourth function, called the third derivative of \(f\text{.}\) And so on.

- 2.15: (Optional) — Is \(\lim_{x\to c}f'(x)\) Equal to \(f'(c)\text{?}\)
- Consider the function \[ f(x) = \begin{cases} \frac{\sin x^2}{x} &\text{if }x\ne 0 \\ 0 &\text{if }x=0 \end{cases} \nonumber \] For any \(x\ne 0\) we can easily use our differentiation rules to find \[ f'(x) = \frac{2x^2\cos x^2 -\sin x^2}{x^2} \nonumber \]