Question

differentiation formulas

Answer 1

In the first section of this chapter we saw the definition of the derivative and we computed a couple of derivatives using the definition.  As we saw in those examples there was a fair amount of work involved in computing the limits and the functions that we worked with were not terribly complicated.

 

For more complex functions using the definition of the derivative would be an almost impossible task.  Luckily for us we won’t have to use the definition terribly often.  We will have to use it on occasion, however we have a large collection of formulas and properties that we can use to simplify our life considerably and will allow us to avoid using the definition whenever possible.

 

We will introduce most of these formulas over the course of the next several sections.  We will start in this section with some of the basic properties and formulas.  We will give the properties and formulas in this section in both “prime” notation and “fraction” notation.

 

Properties

1)                OR          

In other words, to differentiate a sum or difference all we need to do is differentiate the individual terms and then put them back together with the appropriate signs.  Note as well that this property is not limited to two functions.

See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this property.  It’s a very simple proof using the definition of the derivative.

 

2)              OR         ,    c is any number

In other words, we can “factor” a multiplicative constant out of a derivative if we need to.  See theProof of Various Derivative Formulas section of the Extras chapter to see the proof of this property.

 

Note that we have not included formulas for the derivative of products or quotients of two functions here.  The derivative of a product or quotient of two functions is not the product or quotient of the derivatives of the individual pieces.  We will take a look at these in the next section.

 

Next, let’s take a quick look at a couple of basic “computation” formulas that will allow us to actually compute some derivatives.

 

Formulas

1)      If  then       OR      

The derivative of a constant is zero.  See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this formula.

 

2)      If  then           OR      , n is any number.

This formula is sometimes called the power rule.  All we are doing here is bringing the original exponent down in front and multiplying and then subtracting one from the original exponent.

 

Note as well that in order to use this formula n must be a number, it can’t be a variable.  Also note that the base, the x, must be a variable, it can’t be a number.  It will be tempting in some later sections to misuse the Power Rule when we run in some functions where the exponent isn’t a number and/or the base isn’t a variable.

 

See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this formula.  There are actually three different proofs in this section.  The first two restrict the formula to n being an integer because at this point that is all that we can do at this point.  The third proof is for the general rule, but does suppose that you’ve read most of this chapter.

 

These are the only properties and formulas that we’ll give in this section.  Let’s compute some derivatives using these properties.

 

Answer 2

Answer:

Step-by-step explanation:

See the attachment