Derivatives

Finding the derivative is easier than you think. Instead of using Newtons or Leibnitz's equations, in the beginning we can use a simple rule. To derivate, simply use the rule: f'(x) = nx^n-1 .  However, let's look at some simple rules to keep it even easier!

• Rule 1: Constants

  • Let's suppose we have the following equation: f(x) = x^2 + 3. In a case like this, where you have a constant, which in this case is 3, that constant will become 0. So you can ignore it, and leave it out of your derivative function. Hence the derivative of f(x) will be:  2x.

• Rule 2: Coefficients

  • Let's suppose we have the following function: x^2 + 4x + 5. We already know from rule 1 that 5 will fall out. So we can ignore that. However, when it comes to coefficients in front of the variable, (in our case x) simply ignore the x and write down the coefficient only. Hence, our derivative function will be: f'(x) = 2x + 4.  It is very IMPORTANT to notice that if there coefficient of a variable that is not the parameter of the function, both the coefficient and the variable should be ignored! So if instead of f(x) = x^2 + 4x + 5, we can have the function: f(x) = x^2 + 4t+ 5, then you should notice that the t variable is not the functions parameter, hence we can ignore it. So the derived function would be: f'(x) = 2x.

• Rule 3: Multiply that coefficient!

  • Lets suppose that we have the following function: f(x) = 3x^2. You may notice that there is already a coefficient in front of x, and since we'd like to put the value of the exponent in front, that may be a concern for some people. However, the only thing you need to do, is to multiply that value with n. Hence, our derivative function would be: 6x.

And this is everything you need to start differanciation! Have fun derivating!

Let h(x) = 6x^3 + 7x +8. Write down h'(x). 

Multiply n by the coefficient which in this case would be 3 × 6.  Then we subtract one from the exponent. So far it looks like this: 18x^3.

In the case of 7x, we can remove x, so we will keep 7 only. 8 is a constant, so that will become zero. Therefore we can ignore that. 

Therefore, 

h'(x) = 18x + 7  

That is it.