Differentiation 1
What is a derivative?, Polynomials, Stationary Points, Derivatives from First Principles.
Why we study this
Differentiation and its inverse process Integration together comprise Calculus, one of the most important topics covered by advanced maths courses. Differentiation deals with rates of change. The rate of change of displacement is the velocity, and that of velocity is acceleration. Newton found that acceleration is proportional to the forces on an object, and if these are constant we have constant acceleration, and we can calculate velocities and displacements using the constant acceleration formulae. But what if the forces are not constant? For example, the air resistance forces on a car increase as the speed increases - how do we find the velocity and displacement now?
Rates of change apply to many other situations, such as pandemics, population growth, economics, biology, chemistry, ... literally endless applications.
Calculus also plays a crucial role in the evolution of pure mathematical ideas. If we measure angle in radians, the derivative of sin x turns out to be cos x. The rate of change of e^x turns out to be e^x. And if we allow for infinite series, we can find power series for common functions using the ideas of calculus.
_gif.gif)
History of Differentiation
These two remarkably alike bewigged gentlemen are Isaac Newton and Gottfried Leibnitz, the founders of Calculus.
Calculus grew from a long search for better ways to describe change, motion, and shape. Ancient Greek thinkers studied curves and areas, while later mathematicians developed methods that came closer to the ideas of limits and rates of change.
In the seventeenth century, Newton and Leibniz independently developed the main ideas of calculus. Newton focused on motion, gravity, and changing quantities, while Leibniz built a clear system of notation that made differentiation and integration easier to write and use.
Their work led to a fierce priority dispute. Supporters on both sides argued over who had discovered calculus first, and the rivalry became bitter. Historians now recognise that Newton developed his ideas earlier, while Leibniz published first and created the notation used by mathematicians today. Despite the dispute, both men helped shape the calculus we still learn and use.
1665
Newton develops 'method of fluxions'.
1675
Leibniz discovers integral calculus independently.
1684
Leibniz publishes the first work on calculus.
1704
Newton publishes Opticks with calculus methods.
Differentiation Exam Tips
A lot of people find the ideas of calculus quite daunting, but the basic idea of differentiation is quite simple. A straight line has a constant gradient throughout its length - that's why it is straight! The gradient of a curve, on the other hand, varies. throughout its length. Differentiation tells us how the gradient depends on the position on the line. The language and notation can take a bit of getting used to, and proving the results is quite a subtle algebraic process involving the idea of limits (which perhaps explains why it took until the 18th century to fully establish them.
What is a derivative?
The derivative (or differential) of a function of x gives the gradient of the tangent to the graph of f(x) in terms of the x-coordinate. The process of finding the derivative is called differentiation. The derivative of f(x) is written f'(x), and for the graph y = f(x), the derivative is called dy/dx - the origin of this notation becomes clear when we prove results - this process is called differentiation from first principles (which means we don't just use the formula!). The derivative of x^n turns out to be n x^(n-1), as the video below investigates.Â
This formula works for negative and fractional values of n. Don't get these mixed up with integration formulae.
It is important to remember that the derivative represents a rate of change, so if it is positive then the function is increasing with x, and if negative it is decreasing.
We can differentiate polynomials term by term, as the derivative of f(x) + g(x) is just f'(x) + g'(x), and the derivative of kf(x) is just k f'(x). Remember that the derivative of a constant is zero.Â
However, the derivative of a product or quotient of two functions is not the product or quotient of their derivatives - these are dealt with using the product and quotient rules.
Example: Find the derivative of f(x) = x³ − 2x + 5.
Differentiate each term:Â
f(x) = x³ − 2x + 5
f′(x) = 3x² − 2
Derivative of Polynomials
Higher Derivatives
If we differentiate the derivative of a function, we get the second derivative. This is written f"(x) or d²y/dx²
Example: Find the second derivative of f(x) = x³ − 2x + 5.Â
f(x) = x³ − 2x + 5
f′(x) = 3x² − 2
f"(x) = 6xÂ
Repeating this process gives higher derivatives (third, fourth etc).
Stationary points
Calculus gives us a powerful tool for sketching functions. Values of x where the derivative is zero are stationary values, and the coordinates of corresponding points are either local maximums, local minimums or points of inflexion with zero gradients. The video on the left is a reminder.
It's important to note that if both the first and the second derivative are zero for a value of x, we can't conclude that this is a point of inflexion. The second derivative test breaks down here and we need to investigate the sign of the gradient before and after the stationary value.
The video on the right shows the importance of stationary points in sketching functions. The curve looks like y = 2x^2, but if we zoom in we can see it actually has an infinite number of turning points!
​
Derivatives from First Principles
Just investigating the gradient of curves at different x-coordinates doesn't prove anything. Why is the derivative of x^2 equal to 2x? The gif below shows how we can find derivatives from first principles (without quoting formulas). In it, P has x-coordinate a and Q has x-coordinate a+h. The chord joining P and Q is seen to approach the tangent at P as h gets smaller and smaller.
The gradient of the chord is [f(a+h) - f(a)]/h. So the gradient of the tangent must be the limit of this expression as h tends to zero. Click the video below for an example.[text].
Why dy/dx? We call a small change in x δx, and if we write h, the small change in y, δy, then the gradent of the chord becomes δy/δx. So the tangent is the limit of this expression as δx tends to zero, and this is written dy/dx. At the end of the limit process, dx and dy both become zero, so dy/dx becomes 0/0! But the limit exists nevertheless, as we can cancel an 'h' top and bottom algebraically, as in the example. You can see this happen on the gif above, when Q momentarily crashes into P and the chord instantaneously disappears!