Mastering Derivatives: A Comprehensive Guide to Finding Gradients
Derivatives play a fundamental role in calculus and in higher maths in general. They are crucial to understanding the behaviour of functions and how to deal with this. Whether you’re a student learning calculus or a seasoned professional looking to brush off the old manuals, I hope this article will serve as a comprehensive guide to various methods to find derivatives. By exploring different approaches and techniques, you’ll gain a deeper understanding of how derivatives work and how to apply them in different scenarios. They were discovered by Newton or Leibniz, depending on who you ask:
Understanding The Basics
A derivative is in essence a gradient, it helps us extend the concept of a gradient as normally we only consider straight lines to have a gradient. This is right as they are the only type of line that has a constant derivative along them. Curved lines such as that of a quadratic have changing derivatives, this means that we must calculate the formula for the derivative, which we can then use to find the specific case that we want the gradient of.
Imagine you are trying to find the gradient of the graph x² at the point (a, a²). You might start out by drawing a tangent line and using a method line “change in y by change in x”. The problem is, how do we know where to draw the tangent, or how to do this only in algebra?
The key is to use two points on the graph, we can then use these to determine the gradient between the or the gradient of the line that connects them.
You might then decide to find a formula for this.
Then you might ignore the parts that we used to define this try to make r zero, thus making the two points become only one, this still gives us a formula for the gradient. This is then known as the derivative.
Before we meet any more differentiation, we must see the notation.
There we go, we instead of using deltas as we would normally use for a non-zero value, here we have shifted the notation to represent a fraction that doesn’t really make sense as a normal fraction, but the beaty of it is that is almost always behaves like a fraction!
When performing a derivative of a function, as notated we often use d/dx to just mean take the derivative of this next term. And when we notate the derivative of a function we often use something called prime notation:
Differentiation Rules
Power Rule
Power rule states that if you have an equation:
You can calculate the derivative as
This can be said in short as to take the derivative of a function you multiply by the power and subtract one from that same power.
Constant Rule
This rule is that if we have an equation that is a constant, its derivative is zero. Here is an example of me deriving this using the equation we had earlier.
It is a basic one and makes a lot of intuitive sense, if you have a graph y equals five.
It is just a horizontal line, and because it’s y co-ordinates are not changing, it has a gradient at all points of zero.
Sum and difference rules
The derivative is a linear operator.
Let’s unpack that statement. The word linear means that you can share the derivative out over added or subtracted terms making no difference to the end result. Here is an example of using this and the power rule we discovered earlier.
Do you see here how I have shared the derivative and taken the derivative of x³ and x² separately.
Product rule
This is a far more complex one and is far less intuitive. Here it is set out.
We can see an example below of solving the derivative of a multiple of two functions:
This is a very useful rule and can be used in many cases. It appears all the time and is one that you must be able to spot instantly over time.
Quotient rule
This is a version of Product rule, but instead of multiplication it is for division.
This is another one of those must haves, and here is an example of it in use.
Chain Rule
Chain rule is very likely the most important of these basic rules and is able to be used in almost all more complicated expressions. It states that:
Now this might be starting to look like a mass of symbols but bear with me, here is where that fraction thing comes into play.
If we have an equation
and we want to find the derivative of it, we can rename 2x+3 as u, then replacing this in the equation gives us:
We know how to find the derivative of this (with respect to u), it is just:
Now we want dy/dx so how about using our substitution equation to help us out:
We can then multiply these together to get:
What an amazing formula!
Exponential and logarithmic Functions
Now exponentials are a little more difficult and this is because they differentiate to horrible expressions, but we can work them out, just bear with me.
I shall tell you one thing first, which is a nice fact about the exponential function:
It gives itself as a derivative.
We can then use this simple fact to calculate it for any other bases. If we have the expression below, we can simplify this and make it in terms of e just by using logarithm rules:
Here we can then use chain rule to calculate that:
This is a general rule for all exponentials.
Now onto logarithms, when we have a log, we can perform log change of bases just like we did for the exponential to get it in terms of the natural logarithm, this there is then a formula for this:
For this we do not need to use chain rule but one nice fact that comes out of this general formula is that.
Isn’t that a beauty.
There are many other differentiation rules but if you want to learn them, I urge you to go do your own research and escape into the mathematical wilderness of trigonometric functions or other complex equations.
Implicit Differentiation
Implicit differentiation is another method of finding derivatives, but it is more useful for functions that can’t be written as y=f(x). We can separate these into three types.
Functions of x
When calculating the d/dx of a function of x we just find the derivative as we did before, this makes the notation we used above maybe now make sense.
Functions of y
when finding the d/dx of a function of y we take the derivative of the function ignoring that the variable is now y, then we add a multiple of dy/dx at the end. Let’s see an example,
It may look a little but complicated but in reality, it is just another one of those things that results from it being very like a fraction, and it will become second nature after enough time.
Functions of both y and x
When dealing with functions of y and x, we try to use rules we learned about earlier to find the derivative whenever we calculate the derivative of a function only involving y, we add dy/dx on the end. Let’s see an example,
This may be a bit tedious, but it will eventually, end up as often something nice and it is always nice when you bring all of these together.
Finding the derivative of an implicit function
We have the equation for the unit circle.
and we want to find the derivative of it. We must because it can’t be written as a nice function just in terms of x, use implicit differentiation.
Using all the tools we have learned so far, we can do this.
We then solve for dy/dx to find the derivative:
You might have already known this specific example as it is a circle centred at the origin, but it is always nice to watch a what seems like a random process, turn out in such beauty.
Higher order Derivatives
Derivatives are not like normal gradients in one way, we can perform them on a function multiple times, we notate it like this.
As you can see from this example, they work exactly as you would expect them to work. They also have some interesting properties like being able to tell the difference between maxima, minima and points of inflection.
Special Functions
Inverse functions
We can image that the derivative is a fraction again for this, as when we take an inverse function’s derivative it is just the reciprocal of the original function, as shown below:
This is a nice property and does make it far easier when an inverse function is hard to calculate on its own.
Parametric Equations
For parametric equations we can just differentiate them with respect to t. For example,
if we really want, we can then use this to calculate dy/dx using the fraction property again.
This is another, obvious way how taking the derivative to be a fraction can massively help us cut down on computation and make the problem intuitive and easy.
L’Hôpital’s Rule
This is a method of finding the limit. Let’s say you have the function:
and you would like to find the value of it when x is zero. You may think this is a trivial task but when you plug in the values an error pops out.
zero divided by zero! When you might have first learned about functions this is where you would have stopped thinking that the value was undefined and that you can’t compute it’s value. But we are not going to compute its value, we will calculate the limit as x approaches the desired value.
L’hôpital’s rule is a great way of doing this and it tells you then when the top and bottom both approach either zero or infinity, then the limit it the derivative of the top over the derivative of the bottom of the function evaluated at the point. Here is the equation, for our example above:
This is just one of the great uses of derivatives.
Applications of Derivatives
There are many applications of derivatives including optimisation which I have an article on if you want to learn more. You can also use them to find tangent lines, and they are used in many other fields of maths and science all the way from computer science to theoretical physics. They were only invented a couple of hundred years ago and they have spread to be one of the most profound mathematical ideas, and most useful in all areas of maths. Mastering derivatives is a must for anyone who wants to go into any of the mathematical sciences and without it you will look weak and unknowing, this is why I encourage you to set yourself problems and find nice puzzles online or in books that challenge your knowledge as there is always more to learn in such a rich and developing field.
Conclusion
Derivatives are a powerful tool in calculus, providing insights into the behaviour of functions and enabling the solution of a wide range of mathematical problems. By mastering the techniques and methods outlined in this article, you’ll be equipped to find derivatives with confidence and apply them to various real-world scenarios. Remember that practice is key, so make sure to work through plenty of examples to solidify your understanding. With persistence and dedication, you’ll become a master of derivatives and unlock a deeper level of mathematical understanding. And as always,
Have fun and never stop solving.
