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THE DERIVATIVE
Summary
- Derivative of usual functions ................................................................................................... 3
1.1. Constant function ............................................................................................................ 3 1.2. Identity function ࢞ ൌ ሻ࢞ሺࢌ ............................................................................................ 3 1.3. A function at the form ࢞ ^ ................................................................................................ 3 1.4. Exponential function (of the form ࢞ࢇ with ࢇ 0): ...................................................... 5 1.5. Function ࢞ࢋ ...................................................................................................................... 5 1.6. Logarithmic function ࢞ ............................................................................................... 5
- Basic derivation rules .............................................................................................................. 6
2.1. Multiple constant ............................................................................................................ 6 2.2. Addition and subtraction of functions ............................................................................ 6 2.3. Product of functions rule ................................................................................................. 7 2.4. Quotient of functions rule ............................................................................................... 8
- Derivative of composite functions .......................................................................................... 9
How do we recognize a composite function? ............................................................................. 9 3.1. The chain rule .................................................................................................................. 9 3.2. Chain derivatives of usual functions.............................................................................. 10
- Evaluation of the slope of the tangent at one point ............................................................. 12
- Increasing and decreasing functions ..................................................................................... 12
The slope concept usually pertains to straight lines. The definition of a straight line is a function for which the slope is constant. In other words, no matter which point we are looking at, the inclination of a line remains the same. When a function is non‐linear, its slope may vary from one point to the next. We must therefore introduce the notion of derivate which allows us to obtain the slope at all points of these non‐linear functions.
Definition
The derivative of a function f at a point ݔ, written ݂ ′ ሻݔሺ, is given by:
݂^ ′ ሺݔሻ ൌ lim∆௫→݂
if this limit exists.
Graphically, the derivative of a function corresponds to the slope of its tangent line at one specific point. The following illustration allows us to visualise the tangent line (in blue) of a given function at two distinct points. Note that the slope of the tangent line varies from one point to the next. The value of the derivative of a function therefore depends on the point in which we decide to evaluate it. By abuse of language, we often speak of the slope of the function instead of the slope of its tangent line.
Notation
Here, we represent the derivative of a function by a prime symbol. For example, writing ݂ ′ ሻݔሺ represents the derivative of the function ݂ evaluated at point ݔ. Similarly, writing ሺ3 ݔ 2ሻ ′ indicates we are carrying out the derivative of the function 3 ݔ 2. The prime symbol disappears as soon as the derivative has been calculated.
The rule mentioned above applies to all types of exponents (natural, whole, fractional). It is however essential that this exponent is constant. Another rule will need to be studied for exponential functions (of type ࢞ࢇ ). The identity function is a particular case of the functions of form ࢞ ^ (with n = 1) and follows the same derivation rule : ሻݔሺ ′ ࢞ሺ ൌ ^ ሻ ′ ൌ 1 ݔ ଵିଵ^ ൌ 1 ݔ ^ ൌ 1 It is often the case that a function satisfies this form but requires a bit of reformulation before proceeding to the derivative. It is the case of roots (square, cubic, etc.) representing fractional exponents.
Examples
ݔ ൌ ݔ√^ ଵ/ଶ^ ^ ൯ݔ൫√
′ ݔቀ ൌ భమ ቁ^ ′ ൌ ଵ
ଶ ݔ^
ଶ ିݔ^
ଵ/ଶ
ݔ√య^ =ݔ^ ଵ/ଷ^ ݔሺ√య^ ሻ ′ =ݔሺ^ ଵ/ଷ^ ሻ ′^ ൌ^ ଵଷ ݔ^ ሺ^
భ యି ଵሻ^ ൌ ଵଷ ିݔ ଶ/ଷ
Beware of rational functions. For example, the function (^) ௫ଵ ర cannot be differentiated in the same manner as the function ݔ ସ. You must first reformulate the function so that "ݔ" is a numerator, forcing us to change its exponent’s sign.
Examples
ଵ ௫ ర^ ିݔൌ^
௫ ర^ ቁ
′ ൌ ሺݔି ସ^ ሻ ′^ ൌ െ4ݔି ସିଵ^ ൌ െ4ݔି ହ^ ൌ െ (^) ௫ସ ఱ
ଵ ௫ య/మ^ ିݔൌ^
௫ య/మ^ ሻ ′^ ൌ ሺݔି^
య మ (^) ሻ ′ ൌ െ ଷଶ ିݔ
య మି ଵ^ ൌ െ ଷଶ ିݔ
ఱ మ (^) ൌ െ ଷ ଶ௫
ఱమ
Finally, a derivate can greatly be simplified by proceeding first, if possible, to an algebraic simplification.
Example
ݔ ଶ ݔ ଷ^ ݔ√
ݔ ଷ^ ݔ ଵ/ଶ^ ݔ ൌ^
That is how the derivative of ௫^
మ ௫ య^ √௫ is^ greatly^ facilitated^ by^ carrying^ out^ the derivative of ିݔ ଷ/ଶ^.
ݔ ଷ^ ݔ√
′ ൌ ൬ݔି
ଷ ଶ (^) ൰
′ ൌ െ
2 ିݔ^
ଷ ଶିଵ^ ൌ െ^3 2 ିݔ^
ହ ଶ
1.4. An exponential function (of the form ࢞ࢇ with ࢇ 0 ):
It is very easy to confuse the exponential function ܽ ௫^ with a function of the form ݔ ^ since both have exponents. They are, however, quite different. In an exponential function, the exponent is a variable.
Given the exponential function ݂ ܽൌ ሻݔሺ ௫^ where ܽ 0. We have
݂ ′ ሺݔሻ ൌ ሺܽ ௫^ ሻ ′ ܽൌ ݈݊௫^ ሻܽሺ
Examples
ሺ3 ௫^ ሻ ′ ൌ 3 ݈݊ ௫^ ሺ3ሻ
௫ ቇ
′ ൌ ൬
݈݊௫ ൬
1.5. The function ࢞ࢋ
Let the function ݂ ሺݔሻ ൌ ݁ ௫. Then
݂ ′ ሺݔሻ ൌ ሺ݁ ௫^ ሻ ′ ݁ൌ ௫
Here is a special case of the previous rule since the function ݂ ࢞ࢋ ൌ ሻݔሺ is an exponential function with ܽ ݁ ൌ.
Therefore ݂ ′ ࢞ࢋሺ ൌ ሻݔሺ ሻ ′ ݈݊࢞ࢋ ൌ ࢞ࢋ ൌ ሻ݁ሺ ሺ1ሻ ൌ ࢞ࢋ
1.6. The logarithmic function ࢞
Given the logarithmic function ݂ ݈݊ൌ ሻݔሺ ݔ. We have
Example 2
ݔ ଶ^ 8൰
′ ݔ݈݊ሺ ൌ ሻ ′^ െ ሺݔି ଶ^ ሻ ′^ ሺ8ሻ ′
ൌ ଵ௫ െ ሺെ2ݔି ଷ^ ሻ 0
Example 3
′ ൌ ൫3ݔ ଵ/ଶ^ ൯ ′^ ሺ2ݔሻ ′^ െ ሺ8ݔି ଵ^ ሻ ′
భ మ (^) ቁ
′ 2ሺݔሻ ′^ െ 8ሺݔି ଵ^ ሻ ′
భ మ (^) ቁ 2ሺ1ሻ െ 8ሺെݔି ଶ^ ሻ
భ మ (^) 2 8ݔି ଶ
2.3. Product rule
Let ݂ ሻݔሺ and ݃ ሻݔሺ be two functions. Then the derivate of the product
ሺ ݂ሺݔሻ ݃ሺݔሻ ሻ ′ ݂ൌ ′ ݂ ሻݔሺ ݃ሻݔሺ ݃ሻݔሺ ′ ሻݔሺ
We must follow this rule religiously and not succumb to the temptation of writing
݃ሻݔሺ ݂൫ ሻ൯ݔሺ ′^ ݂ ൌ ′ ݃ሻݔሺ ′ ;ሻݔሺ a faulty statement.
Example 1
ݔሺ ݁ଷ^ ௫^ ሻ ′^ ݔሺ ൌ ଷ^ ሻ ′^ ݁ ௫^ ݔ ଷ^ ݁ሺ ௫^ ሻ ′
ݔ3 ൌ ݁ଶ^ ௫^ ݔ ݁ଷ^ ௫
Example 2
(ሻݔ݈݊ݔ3√ ′ ൌ ሺ3√ݔሻ ′ ݈݊ ݔ 3 (^) ሻݔ݈݊ሺݔ√ ′
భ మ (^) ቁ
′ ݈݊ ݔ 3 (^) ݔ݈݊ሺݔ√ ሻ ′
భ మି ଵ^ 3 ݔ ݈݊ቁ (^) ݔ√ ଵ௫
భ ݈݊మ (^) ݔ 3ݔି
భ మ
2.4. Quotient rule
Let ݂ ሻݔሺ and ݃ ሻݔሺ be two functions. Then the derivative of the quotient
݂ቆ
′ ݂ൌ
Just as with the product rule, the quotient rule must religiously be respected.
Example 1
′ ൌ
ݔሺ ଷ^ ሻ ′ ݁^ ௫^ ݔ െ ଷ^ ݁ሺ ௫^ ሻ ′
݁ሺ ௫^ ሻଶ
ൌ ଷ௫^
మ (^) ೣି ௫ య (^) ೣ ሺೣ ሻ మ
ൌ ௫^
మ (^) ೣ ሺଷି௫ሻ ሺೣ ሻ మ
ൌ ௫^
మ (^) ሺଷି௫ሻ ೣ
Example 2
′ ൌ
൫3√ݔ൯ ′ ݈݊^ ݔെ 3 ሻݔ݈݊ሺݔ√ ′
ሻݔ݈݊ሺ ଶ^ ൌ
ଵ ଶ (^) ൰
′ ݈݊ ݔെ 3 (^) ሻݔ݈݊ሺݔ√ ′ ሻݔ݈݊ሺ ଶ
ଷቆభమ ௫భమషభ^ ቇ௫ିଷ (^) √௫ భೣ ሺ௫ሻ మ^ =
ଷ௫ షభమ^ ௫ି௫ షభమ ଶሺ௫ሻ మ
ൌ ଷ௫^
షభమ (^) ሺ௫ିଶሻ ଶሺ௫ሻమ
and then multiplying it with the derivative of the internal function. If the latter is also composite, the process is repeated. Be alert as the internal function could also be a product, a quotient, ...!
3.2. Chain derivatives of usual functions
In concrete terms, we can express the chain rule for the most important functions as follows :
If ሻݔሺ ݃ൌ ݑ represents any given function of x
ݑሺ ^ ሻ ′ ݊ ൌ ݑ ିଵ^ ݑ ′ ܽሺ ௨^ ሻ ′ ܽ ൌ ݈݊ ௨^ ݑሻܽሺ ′ ݁ሺ ௨^ ሻ ′ ݁ ൌ ௨^ ݑ ′ ݑ ݈݊ሺ ሻ ′^ ൌ ଵ௨ ݑ ൈ ′
Examples
ሾlnሺݔ ଶ^ 2 ݔ 1ሻሿ ′^ ൌ
ݔ ଶ^ 2 ݔ 1 ݔሺ^
݁െ ݔ3 ݔ݈݊ሾሺ ௫^ ሻସ^ ሿ ′^ ݁െ ݔ3 ݔ݈݊ൌ 4ሺ ௫^ ሻଷ^ ݁െ ݔ3 ݔ݈݊ሺ ௫^ ሻ ′
݁െ ݔ3 ݔ݈݊ൌ 4ሺ ௫^ ሻଷ^ ሺ ଵ௫ 3 െ݁ ௫^ ሻ
݁ሺ ଷ௫ିହ^ ሻ ′ ݁ൌ ଷ௫ିହ^ ሺ3 ݔെ 5ሻ ′
݁ൌ ଷ௫ିହ^. 3
Below are additional examples that demonstrate that many rules may be necessary for one derivative.
Example 1
ሺሾln ሺ3ݔ ଷ^ ݁9 െ ௫^ ሿଷ^ ሻ ′^ ൌ 3ሾln ሺ3ݔ ଷ^ ݁9 െ ௫^ ሻሿଶ^. ሾln ሺ3ݔ ଷ^ ݁9 െ ௫^ ሻሿ ′
ൌ 3ሾlnሺ3ݔ ଷ^ ݁9 െ ௫^ ሻሿଶ^.
ݔ3 ଷ^ ݁9 െ ௫^. ሺ3ݔ^
ൌ 3ሾlnሺ3ݔ ଷ^ ݁9 െ ௫^ ሻሿଶ^.
ݔ3 ଷ^ ݁9 െ ௫^. ሺ9ݔ^
Example 2
݁ሾ ௫௫^ ሿ ′^ ݁ൌ ௫௫^ ݔ݈݊ݔሺ. ሻ ′
݁ൌ ௫௫^ ሻݔሾሺ ′^ ݈݊ݔ ݔ ݔ݈݊ሺ ሻ ′ ሿ^ (product rule)
݁ൌ ௫௫^ .ݔ ݔ ݈݊.ቀ1 (^) ௫ଵ ቁ
݁ൌ ௫௫^ 1 ݔ݈݊ሺ ሻ
Example 3
ቈ ௫^
మ (^) ାଵ ሺଶ௫ାଵሻ భమ
′ ൌ
൫௫ మ^ ାଵ൯ ′ .ሺଶ௫ାଵሻ భమି^ ൫௫ మ^ ାଵ൯.ቈሺଶ௫ାଵሻ భమ^
′
ቈሺଶ௫ାଵሻ భమ^
మ (quotient^ rule)
ଶ௫.ሺଶ௫ାଵሻ భమି^ ൫௫ మ^ ାଵ൯.ቈሺଶ௫ାଵሻ భమ^
′
ଶ௫ାଵ
ଶ௫.ሺଶ௫ାଵሻ
భమି ൫௫ మ^ ାଵ൯. భమ ሺଶ௫ାଵሻ ష
భమ .ሺଶ௫ାଵሻ ′ ଶ௫ାଵ
భమି (^) ൫௫ మ^ ାଵ൯. భమ ሺଶ௫ାଵሻ షభమ (^) .ଶ ଶ௫ାଵ
ൌ ଶ௫.ሺଶ௫ାଵሻ^
భమି ൫௫ మ^ ାଵ൯.ሺଶ௫ାଵሻ ష
భమ ଶ௫ାଵ
Solution
We need to derive the composite function ݑ ଷ, where ݔ ൌ ݑ ଶ^ െ 4. Consequently, we need to use the chain derivative.
݂^ ′ ሺݔሻ ൌ ሾሺݔ ଶ^ – 4ሻ ଷ^ ሿ ′
ൌ 3ሺݔ ଶ^ – 4ሻ ଶ^ ݔሺ. ଶ^ – 4ሻ ′
ൌ 3ሺݔ ଶ^ – 4ሻ ଶ^ ݔ.
ݔሺݔ6 ൌ ଶ^ – 4ሻ ଶ
At point ݔൌ 1, the slope of the tangent of the function ݂ is
݂ ′ ሺ1ሻ ൌ 6ሺ1ሻሺ1 ଶ^ – 4ሻ ଶ^ ൌ 6ሺ1ሻሺ – 3ሻଶ^ ൌ 54
Since the slope is positive at ݔൌ 1, the function ݂ ሻݔሺ is increasing at this point.
The slope is 0 at points like ݂ ′ ሺݔሻ ൌ 0. We therefore need to find the values of x so that
ݔሺݔ6 ଶ^ – 4ሻ ଶ^ ൌ 0
ݔൌ 0, ݔൌ െ2 and ݔൌ 2 are the values sought.
