Calculus For Everyone

Derivative Formula Of A Function

Standard notation, derivative of some common functions, and chain rule for differentiation

As I explained in the previous article, the main subject of derivatives/differentials is about how fast a change occurs. And — as you know — the rate of change is represented by the gradient of the function on the graph. That is, the greater the gradient of a function, the faster the value of the function changes.

In the previous article, we have seen that the gradient of a function at a point can be expressed as f’(x). f’(x) also shows the rate of change of the function f(x). f’(x) is what we called the derivative/differentiation of f(x).

Notation

Before proceeding to the next discussion, you should know that there is a special notation that can replace the formula for the derivative above. ∆x is denoted as dx, and f(x+∆x)-f(x) is denoted as df(x) or dy. the letter d in dx and dy represents a change in value, replacing the symbol ∆x. the use of this notation with the letter d also results in writing limit x->0 is no longer needed because the letter d in dx itself represents a very small change from x. So, we can say that f’(x)= dy/dx. This notation means "divide a very small value of y by a very small value of x".

This notation f’(x) = dy/dx is what we will often use in the differential later.

Derivatives Of Some Common Functions

There are several derivatives of general functions that you should keep in mind. The function is f(x) = ax^n ( a and n are constants ), f(x) = sin x, f(x) = cos x, f(x) = e^x ( e is a constant known as euler’s number ), and f(x) = ln x.

1. Derivative Of f(x) = ax^n

From the above calculations, we can conclude that the derivative of ax^n is anx^(n-1). For example, f(x) = 3x⁵, then the derivative — f’(x) — is 3×5x⁴=15x⁴. From this formula, we also know that the derivative of a constant is 0. So, the derivative of 3 is 0 and the derivative of 4 is also 0.

2. Derivative Of f(x) = sin x And f(x) = cos x

From the above calculation, we can see that the derivative of sin x is cos x and the derivative of cos x is sin x.

3. Derivative of f(x) = e^x

From the description above, we can see that the function f(x)=e^x is very unique. No matter how many times it is differentiate, the result is still e^x.

4. Derivative Of f(x) = ln x

From the above calculation, we know that the derivative of the ln x is 1/x.

Chain Rule For Differentiation

With the above formula, it is very easy for us to calculate the derivative of a function. For example, the derivative of x² can be easily calculated. But what about the derivative of (x+3)²? It’s easy, we just need to expand it by calculating (x+3)(x+3) as usual. What about the derivative of (x+3)⁵⁶? Do you still want to expand it? No, you will waste too much time. Luckily, we have a chain rule to solve this.

Here’s a statement I quoted from khanacademy.org :

The chain rule states that the derivative of f(g(x)) is f’(g(x))⋅g’(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x².

So, the chain rule can be used to derive the composite function. In case you forgot, a composite function is simply a function in which there is another function. The general form is f(g(x)), where the function g(x) becomes the domain of the function f(x). An example is the function (x+3)⁵⁶ earlier. The function g(x) here is g(x)=x+3 and f(x) =x⁵⁶. Then, how to use the chain rule for composite function derivatives? It’s easy. Suppose the function you want to derive is f(g(x)). To calculate the derivative, you must derive the function f(g(x)) with respect to g(x), then multiply by the derivative of the function g(x) with respect to x. For the function f(g(h(x))), the method is the same. Derive the function f(g(h(x))) with respect to g(h(x)), then multiply the result by the derivative of g(h(x)) with respect to h(x). Then the result is multiplied by the derivative of h(x) with respect to x. This rule applies because basically, because the derivative (dy/dx) is just a fraction, so we can break it down into smaller fractions like this :

Example

What is the derivative of (x+3)⁵⁶?

Exercise

What is the derivative of the following function f(x)?

Enough for today. You can give a clap if you find this article useful and helpful. Oh yes, to read this Calculus For Everyone series from the beginning, click here.

Thank you for reading this article!