# Introduction To Derivative Formula, Introduction To Derivatives

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.

Đang xem: Derivative formula

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. Turunan dari 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.