Using the derivative formula to find the derivative of a function is a hassle. There are many shortcuts to finding the derivative. The first one is the Power Rule. It can be used with functions which are equal to a polynomial like . Each term’s coefficient is multiplied by the exponent and the exponent decreases by one. Let’s look at . The derivative’s coefficient will be 6 because the coefficient of the term, 3, is multiplied by the exponent, 2. The exponent will be 1 because the exponent is decreased by one. Therefore, the derivative is . If the function has more than one term, use the same procedure for each term. Now let’s use the power rule to find the derivative of . The derivative is . It is important to notice that the constant, -2, is eliminated since x has an exponent of zero. Also, 3x becomes 3 since anything raised to the power of 0 is equal to 1.
Another useful shortcut is the Product Rule. It is useful when the definition is two polynomials multiplied by each other but cannot easily be simplified such as . If u is assigned to the first polynomial and v is assigned to the second polynomial, the derivative is . So in and . Using the power rule, and . Therefore, by using the product rule .
The last rule I am going to discuss for now is the Quotient Rule. The Quotient Rule is used when a polynomial is being divided by another polynomial. Where , . For example, let . The u is and v is . Once again utilizing the power rule, we find that and . Plugging these values into the quotient rule yeilds .
Proper use of these rules save time. On some occasions, multiple rules will need to be applied. There is another rule I have not yet mentioned called the chain rule. It makes taking the derivative of a function within a function possible. That post will be saved for after the explanation of the usefulness of derivatives.