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Shortcuts for Finding Derivatives

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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 x^3 + 5x^2 - 7. Each term’s coefficient is multiplied by the exponent and the exponent decreases by one. Let’s look at f(x)=3x^2. 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 f'(x)=6x. 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 g(x)=2x^4-5x^2+3x-2. The derivative is g'(x)=8x^3-10x+3. 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 f(x) = (x^2 - 2)(3x^2 + x - 4). If u is assigned to the first polynomial and v is assigned to the second polynomial, the derivative is u\frac{dv}{dx}+v\frac{du}{dx}. So in f(x) u = x^2 - 2 and v = 3x^2 + x - 4. Using the power rule, \frac{du}{dx} = 2x and \frac{dv}{dx} = 6x + 1. Therefore, by using the product rule f'(x) = (x^2 - 2)(6x + 1) + (3x^2 + x - 4)(2x) = 12x^3 + 3x^2 - 20x - 2.


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 y=\frac{u}{v}, \frac{dy}{dx} = \frac{ v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}. For example, let f(x)=\frac{x^2+2}{x-1}. The u is x^2 + 2 and v is x - 1. Once again utilizing the power rule, we find that \frac{du}{dx} = 2x and \frac{dv}{dx} = 1. Plugging these values into the quotient rule yeilds f'(x) = \frac{(x-1)(2x)-(x^2+2)(1)}{(x-1)^2} = \frac{x^2-2x-2}{x^2-2x+1}.


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.

Written by todizzle91

August 13th, 2009 at 10:35 am

Posted in Calculus

3 Responses to 'Shortcuts for Finding Derivatives'

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  1. plz update me with limit and derivatives


    21 Mar 10 at 2:10 am

  2. You are a cool young man. Thank you for explaining this so clearly.

    Juan Wiltocruz

    20 May 11 at 9:58 am

  3. This was really helpful, especially when my math teacher did properly explain each rule. Thank you. And now I know where to come, if he does it again.


    28 Nov 12 at 3:21 am

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