Introduction to Derivation Rules

Introduction to Derivation RulesBy Jason Lam

ƒ´(x)= ddxƒ(x)

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MENUWhat would you like to derive?

Learn how constant functions behave when derived.

Learn how to derive functions that have a variable to any power.

These are some very simple rules that do not fall under the above categories. They are very useful and necessary for further progress.

Challenge your understanding of the above content with this 10 questions quiz.

Constant Functions

Power Functions

Quiz

Exponent Functions

Learn how to derive functions when the exponent is a variable.

Extra Rules

Constant Functions

𝑓 ′ (𝑥 )=limh→0

𝑓 (𝑥+h )− 𝑓 (𝑥 )h

=limh→0

𝑐−𝑐h

=¿ limh→0

0=0¿

The below proof shows that for a function f(x) where f(x) is a constant value c, the slope is 0. We can also say f’(x) = 0.

Constant FunctionsSo the derivative of a constant function where c is a constant

value is as follows:

𝑑𝑦𝑑𝑥 (𝑐 )=0

𝑦= 𝑓 (𝑥 )=𝑐𝑑𝑦𝑑𝑥= 𝑓 ′ (𝑥 )=0

Slope = 0

Constant FunctionsFor example, given a function such that y = 127, what is dy/dx?

So, the derivative of any constant is zero. This also means that the slope of any constant is zero.

𝑦=127 ¿𝑑𝑑𝑥 127

¿0

𝑑𝑑𝑥 𝑦 The derivative of y

Substitute for y

Apply constant rule

Constant FunctionsYou have now learned how to derive constant functions. Click

“Menu” for more options or “Restart” to go over constant functions once more.

GOOD JOB!

Power Functions

Where “n” is any real number.

Power functions can be derived using the

Power Rule:

Power Functions

In this case: x = x n = 6

Given what is f’(x)?

Following the power rule, identify x and n:

𝑑𝑦𝑑𝑥 =

𝑑𝑑𝑥 (𝑥6)

¿6 𝑥6−1

¿6 𝑥5

Power Functions

In this case: x = x n = -1

The power rule still applies even if the power is negative. Given find ?

Rewrite the equation so that

𝑑𝑦𝑑𝑥=

𝑑𝑑𝑥 (𝑥−1)

¿−1 𝑥− 1−1 or

y= 1𝑥

Power Functions

In this case: x = x n =

The power rule still applies even if the power is a fraction. Given find ? We use the same technique from the previous slide.

Rewrite the equation so that

𝑑𝑦𝑑𝑥 = 𝑑

𝑑𝑥 (𝑥12 )

¿ 12𝑥

( 12 −1)

or

y=√𝑥

Power FunctionsYou have now learned how to derive constant functions. Click

“Menu” for more options or “Restart” to go over power functions once more.

EXCELLENT!

Exponential Functions

This is true for all exponential functions but for now le us work with the function

The rate of change of any exponential function is proportional to the function itself. In other words, the slope is proportional to the height.

Exponential Functions

S is its own derivative. How cool is that?

The derivative of the natural exponential function is as follows:

𝑑𝑑𝑥 𝑒

𝑥=𝑒𝑥

Exponential Functions

Because of this unique property, the following is true:

𝑓 (𝑥 )=𝑒𝑥

𝑓 ′ (𝑥 )=𝑒𝑥

𝑓 ′ ′ (𝑥 )=𝑒𝑥

𝑓 ′ ′ ′ (𝑥 )=𝑒𝑥

::And so on

y=𝑒𝑥

Exponential Functions

can be used with various other functions. For example, given find and using the constant and power rule we get:

Exponential FunctionsYou have now learned how to derive constant functions. Click

“Menu” for more options or “Restart” to go over exponential functions once more.

FANTASTIC!

Extra RulesHere you will learn the constant multiple rule, sum rule and difference rule. They are all very easy to understand will help you derive many variations to common functions.

Extra Rules

Constant Multiple Rule states that

𝑑𝑑𝑥 [𝐶𝑓 (𝑥 ) ]=𝐶 𝑑

𝑑𝑥 𝑓 (𝑥)

Where C is a constant real number and f(x) is a differentiable equation.

Extra Rules

So given find .We can find this using the power and constant multiple rule:

𝑓 ′ (𝑥 )=𝑑𝑑𝑥 (3 𝑥4 )

¿3 𝑑𝑑𝑥 (𝑥4)

¿3 (4)(𝑥4−1)¿12 𝑥3

Find derivative of the given function.

First move the constants to the outside of the derivative.

Next you can apply the power rule to the given variable.

Finally we have the answer!

Extra RulesThe Sum/Difference Rule states that the derivative of a sum/difference of functions is the sum/difference of the individual derivatives. SO:

Where f(x) and g(x) are differentiable functions.

Extra Rules

So given find .We can find this using the power and sum/difference rule:

𝑓 ′ (𝑥 )=𝑑𝑑𝑥 𝑓 (𝑥)

¿ (8 ) (𝑥8− 1)+(12 ) (5 ) (𝑥5−1 )− (6 ) (1 ) (𝑥1−1 )+0¿8 𝑥7+60 𝑥4−6

Extra RulesYou have now learned the Extra Rules part of this lesson. Click

“Menu” for more options or “Restart” to go over the extra rules once more.

AWESOME!

Quiz IntroductionUse the techniques and rules from this lesson to successfully

answer all 10 of the following questions. See if you can get 8 of the 10 correct!

QUIZ TIME!GOOD LUCK!

Question 1Given what is f’(x)?

BA

CD

18.65

186.5

x

0

Great Job!

Correct! Using the constant rule We know that f’(x) = 0 if f’(x) is a constant value equal to 186.5

Ooooops!That was incorrect, would you like to try again?

HINT:

Remember what the constant rule tells us about deriving constant values?

𝑑𝑦𝑑𝑥 (𝑐 )=0

Click the X button for no.Click the checkmark for yes.

Question 2Given what is f’(x)?

BA

CD

1/3

1/30

15

0

Great Job!

Correct! Using the constant rule We know that f’(x) = 0 if f’(x) is a constant value equal to

Ooooops!That was incorrect, would you like to try again?

HINT:

Remember what the constant rule tells us about deriving constant values?

𝑑𝑦𝑑𝑥 (𝑐 )=0

Click the X button for no.Click the checkmark for yes.

Question 3Differentiate the function

BA

CD

6 𝑥7

8 𝑥7

34 𝑥

7

0

Great Job!Correct! Using the power rule and constant multiple rule:

Ooooops!That was incorrect, would you like to try again?

HINT:

Remember what the power rule and the constant multiple rule. Start here:

Click the X button for no.Click the checkmark for yes.

𝑑𝑑𝑥 ( 34 𝑥8)=( 34 ) 𝑑

𝑑𝑥 (𝑥8 )

Question 4Given find f’(x)

BA

CD

23

2-t

2+t

− 23

Great Job!Correct! Using the sum/difference rule and power rule:

Ooooops!That was incorrect, would you like to try again?

HINT:

Remember what the sum/difference rule states that:

Click the X button for no.Click the checkmark for yes.

Question 5Given find f’(x)

BA

CD

52 𝑥

4

0

5 𝑥4𝑥5

Great Job!Correct! Using the power and constant multiple rule and power

rule:

Ooooops!That was incorrect, would you like to try again?

HINT:

Combine the power rule and the constant multiple rule.

Click the X button for no.Click the checkmark for yes.

𝑑𝑑𝑥 [𝐶𝑓 (𝑥 ) ]=𝐶 𝑑

𝑑𝑥 𝑓 (𝑥)

Question 6Given find v’(r)

BA

CD

43 𝜋

2𝑟

4𝜋𝑟 2

34 𝜋

12𝜋𝑟2

Great Job!Correct! Using the power and constant multiple rule and power

rule:

Ooooops!That was incorrect, would you like to try again?

HINT:

Combine the power rule and the constant multiple rule. Also keep in mind that is a constant value.

Click the X button for no.Click the checkmark for yes.

𝑑𝑑𝑥 [𝐶𝑓 (𝑥 ) ]=𝐶 𝑑

𝑑𝑥 𝑓 (𝑥)

Question 7Given find B’(x)

BA

CD

𝑒𝑥

𝑒𝑥

𝑒𝑥

𝑒−5𝑥

Great Job!Correct! Using the sum/difference rule and the properties of an

exponential function we get:

Ooooops!That was incorrect, would you like to try again?

HINT:

Use the sum/difference rule along with the properties of the natural log e.

Click the X button for no.Click the checkmark for yes.

Question 8Given find v’(r)

BA

CD

−60 𝑥12𝑥4

60 𝑥460 𝑥−6

Great Job!Correct! Using the power and constant multiple rule and power

rule:

Ooooops!That was incorrect, would you like to try again?

HINT:

Rewrite the original function so that it looks like the one below. Also, keep in mind the power rule:

Click the X button for no.Click the checkmark for yes.

𝑓 (𝑥 )=−12𝑥−5

Question 9Given find dy/dx

BA

CD

𝑒𝑥+3

5𝑒𝑥+3

5𝑒𝑥

𝑒𝑥

Great Job!Correct! Using the power and constant multiple rule and power

rule:

Ooooops!That was incorrect, would you like to try again?

HINT:

Remember what the sum rule states, as well as the derivative property of the natural log e.

Click the X button for no.Click the checkmark for yes.

𝑑𝑑𝑥 𝑒

𝑥=𝑒𝑥

Question 10Given a constant “c”, and find B’(x)

BA

CD

6𝑐 𝑥− 6

−6 𝑐 𝑥− 7

−6 𝑐 𝑥5−6 𝑥5

Great Job!Correct! Using the power and constant multiple rule and power

rule:

Ooooops!That was incorrect, would you like to try again?

HINT:

Remember what the sum rule states, as well as the derivative property of the natural log e.

Click the X button for no.Click the checkmark for yes.

𝑑𝑑𝑥 𝑒

𝑥=𝑒𝑥

CONGRATULATIONS!!!Great job! You’ve successfully completed this lesson and quiz!

How did you do on the quiz? Did you get at least 8 right?

You can retake the lesson or quiz if you’d like. Just click menu.