A Definitive Guide to Differentiation

Overview: Differentiation

Differentiation is a very useful mathematical technique extensively applied in various fields including economics, science, technology and engineering. Examples range from obtaining the speed of a space rocket given its acceleration profile to being able to study how hot an object gets if heated. 

In this Q&A, we will be dealing with single-variable functions and good knowledge of algebra is assumed.

The Basics of Differentiation

Given a function, f(x) = ax^n, where a is a constant, x is the variable and n is the exponent:

The derivative of f(x) is df/dx = f '(x) = nax^(n-1).

Note that the derivative of f(x) can also be written as df(x)/dx or df/dx or f'(x). In our case (single-variable functions), either of these are acceptable and it is all down to preference. For ease and neatness, we will be using f'(x). 

The constant term is multiplied by the exponent and the variable will have a new exponent that is less than 1 than its original one.

Applying this to another function, this time with a defined 'a' and 'n'.

Derive f(x) = 3x^2. 

Here a = 3 and n = 2. The variable of the function is x.

Therefore, multiply the constant term, a, with the exponent... 

so that is 3 x 2 = 6

Then subtract the original exponent by 1...

so that is 2 -1 = 1

Therefore, the derivative of f(x) = 3x^2 is:

df/dx = f '(x) = 6x^1. (or simply f '(x) = 6x).

Try to differentiate this function, f(x) = 100x^100.

Your answer should be f '(x) = 10000x^99.

Differentiation of Polynomials

Differentiation of polynomials is not particularly difficult. All you have to do is differentiate each term of the polynomial separately. 

Given a function, f(x) = 2x + 3x^2 + 5x^3.

All you need to do is differentiate 2x separately, then 3x^2, and lastly 5x^3. If we apply the concept learn from before:

The derivative of 2x is 2. The derivative of 3x^2 is 6x. The derivative of 5x^3 is 15x^2. 

Therefore, the derivative of the polynomial function is:

f(x) = 2x + 3x^2 + 5x^3.

f '(x) = 2 + 6x + 15x^2.

Differentiation of Constants

How do we differentiate a number by itself? For example, what is the derivative of 3?

We know that a variable raised to the power of zero is equals to 1. 

Therefore, if we have a number such as 3, we can also say that it is equals to 3x^0, because x^0 = 1.

x^0 = 1

3(1) = 1

3x^0 = 3(1) = 3

Thus, if we differentiate 3, we are differentiating 3x^0.

f(x) = 3x^0. So... f '(x) = 0. Since you are multiplying the constant term, 3 with zero which wipes the whole term away.

This is true for all constants.

The derivative of 1 is zero.

Ther derivative of zero is zero.

The derivative of 24711093 is zero.

Try this: what is the derivative of the number of characters of all the letters in your screen?

Differentiation of a term with a variable in the denominator

We know from algebra that 3/x^2 is equals to 3x^-2.

It is the same strategy. Multiply the constant term by the exponent and subtract the original exponent by 1.

Multiply the constant term 3 with -2: (3)(-2) = -6

Subtract the original exponent: -2 - 1 = -3

So, the derivative of 3x^-2 is -6x^-3

Final words

If you are to take away something from this Q&A, it is this statement:

When deriving a term, multiply the constant term with the exponent, then subtract the original exponent by one.

I hope this helps.