Introduction to Rules for Calculating Derivatives
Calculus is a powerful tool, for much of the physical world that we wish to analyze is best understood in terms of rates of change. It becomes even more powerful when we can find some simple rules that enable us to calculate derivatives quickly and easily. This section is devoted to that topic.
Derivative of a Sum [The Sum Rule]:
We calculate the derivative of a sum (or difference) by
( f ( x ) ± g ( x ))′ = f′ ( x ) ± g′ ( x ).
In our many examples, we have used this fact implicitly. We are now just enunciating it formally.
Derivative of a Product [The Product Rule]:
We calculate the derivative of a product by
[ f ( x ) · g ( x )]′ = f′ ( x ) · g ( x ) + f ( x ) · g′ ( x ).
We urge the reader to test this formula on functions that we have worked with before. It has a surprising form. Note in particular that it is not the case that [ f ( x ) · g ( x )]′ = f ′( x ) · g′ ( x ).
Derivative of a Quotient [The Quotient Rule]:
We calculate the derivative of a quotient by
In fact one can derive this new formula by applying the product formula to g ( x ) · [ f ( x )/ g ( x )]. We leave the details for the interested reader.
Derivative of a Composition [The Chain Rule]:
We calculate the derivative of a composition by
[ f ο g ( x )]′ = f′ ( g ( x )) · g′ ( x ).
To make optimum use of these four new formulas, we need a library of functions to which to apply them.
A Derivatives of Powers x:
If f ( x ) = x ^{k} then f′ ( x ) = k · x ^{k −1} , where k {0, 1, 2, ...}.
Math Note: If you glance back at the examples we have done, you will notice that we have already calculated that the derivative of x is 1, the derivative of x ^{2} is 2 x , and the derivative of x ^{3} is 3 x ^{2} . The rule just enunciated is a generalization of these facts, and is established in just the same way.
B Derivatives of Trigonometric Functions:
The rules for differentiating sine and cosine are simple and elegant:
We can find the derivatives of the other trigonometric functions by using these two facts together with the quotient rule from above:
Similarly we have
Derivatives of In x and e ^{x}:
We conclude our library of derivatives of basic functions with
and
We may apply the Chain Rule to obtain the following particularly useful generalization of this logarithmic derivative:
Now it is time to learn to differentiate the functions that we will commonly encounter in our work. We do so by applying the rules for sums, products, quotients, and compositions to the formulas for the derivatives of the elementary functions. Practice is the essential tool in mastery of these ideas. Be sure to do all the You Try It problems in this section.
Examples
Example 1
Calculate the derivative
Solution 1
We know that ( d/dx ) sin x = cos x , ( d/dx ) x = 1, ( d/dx ) x ^{3} = 3 x ^{2} , and ( d/dx ) In x = (1/ x ). Therefore, by the addition rule,
and
Now we may conclude the calculation by applying the product rule:
Example 2
Calculate the derivative
Solution 2
We know that ( d/dx ) e ^{x} = e ^{x} , ( d/dx ) x = 1, ( d/dx ) sin x = cos x , and ( d/dx ) tan x = sec ^{2} x . By the product rule,
Therefore, by the quotient rule,
You Try It: Calculate the derivative .

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